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Theorem ptcnplem 21234

Description: Lemma for ptcnp 21235. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

**Hypotheses**
Ref	Expression
ptcnp.2	⊢ 𝐾 = (∏_t‘𝐹)
ptcnp.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ptcnp.4	⊢ (𝜑 → 𝐼 ∈ 𝑉)
ptcnp.5	⊢ (𝜑 → 𝐹:𝐼⟶Top)
ptcnp.6	⊢ (𝜑 → 𝐷 ∈ 𝑋)
ptcnp.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))
ptcnplem.1	⊢ Ⅎ𝑘𝜓
ptcnplem.2	⊢ ((𝜑 ∧ 𝜓) → 𝐺 Fn 𝐼)
ptcnplem.3	⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ (𝐹‘𝑘))
ptcnplem.4	⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ Fin)
ptcnplem.5	⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → (𝐺‘𝑘) = ∪ (𝐹‘𝑘))
ptcnplem.6	⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))

**Assertion**
Ref	Expression
ptcnplem	⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))

Distinct variable groups: 𝑧,𝐴 𝑥,𝑘,𝑧,𝐷 𝑘,𝐼,𝑥,𝑧 𝑥,𝐺,𝑧 𝑘,𝐽,𝑧 𝑧,𝐾 𝜑,𝑘,𝑥,𝑧 𝑘,𝐹,𝑥,𝑧 𝑘,𝑉,𝑥 𝑘,𝑊,𝑧 𝑘,𝑋,𝑥,𝑧 Allowed substitution hints: 𝜓(𝑥,𝑧,𝑘) 𝐴(𝑥,𝑘) 𝐺(𝑘) 𝐽(𝑥) 𝐾(𝑥,𝑘) 𝑉(𝑧) 𝑊(𝑥)

Proof of Theorem ptcnplem Dummy variables 𝑓 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcnplem.4	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ Fin)
2		inss2 3796	. . . 4 ⊢ (𝐼 ∩ 𝑊) ⊆ 𝑊
3		ssfi 8065	. . . 4 ⊢ ((𝑊 ∈ Fin ∧ (𝐼 ∩ 𝑊) ⊆ 𝑊) → (𝐼 ∩ 𝑊) ∈ Fin)
4	1, 2, 3	sylancl 693	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝐼 ∩ 𝑊) ∈ Fin)
5		nfv 1830	. . . . 5 ⊢ Ⅎ𝑘𝜑
6		ptcnplem.1	. . . . 5 ⊢ Ⅎ𝑘𝜓
7	5, 6	nfan 1816	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝜓)
8		inss1 3795	. . . . . . 7 ⊢ (𝐼 ∩ 𝑊) ⊆ 𝐼
9	8	sseli 3564	. . . . . 6 ⊢ (𝑘 ∈ (𝐼 ∩ 𝑊) → 𝑘 ∈ 𝐼)
10		ptcnp.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))
11	10	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷))
12		ptcnplem.3	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ (𝐹‘𝑘))
13		ptcnp.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
14	13	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ 𝑋)
15		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
16		ptcnp.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
17	16	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋))
18		ptcnp.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹:𝐼⟶Top)
19	18	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top)
20		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘)
21	20	toptopon 20548	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
22	19, 21	sylib 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)))
23		cnpf2 20864	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
24	17, 22, 10, 23	syl3anc 1318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
25		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
26	25	fmpt 6289	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘))
27	24, 26	sylibr 223	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘))
28	27	r19.21bi 2916	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘))
29	25	fvmpt2 6200	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ (𝐹‘𝑘)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
30	15, 28, 29	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
31	30	an32s 842	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
32	31	mpteq2dva 4672	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = (𝑘 ∈ 𝐼 ↦ 𝐴))
33		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		ptcnp.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉)
36		mptexg 6389	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑉 → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V)
37	35, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V)
38		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))
39	38	fvmpt2 6200	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) = (𝑘 ∈ 𝐼 ↦ 𝐴))
40	33, 37, 39	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) = (𝑘 ∈ 𝐼 ↦ 𝐴))
41	32, 40	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥))
42	41	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥))
43	42	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥))
44		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐼
45		nffvmpt1 6111	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)
46	44, 45	nfmpt 4674	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷))
47		nffvmpt1 6111	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷)
48	46, 47	nfeq 2762	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷)
49		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷))
50	49	mpteq2dv 4673	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐷 → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)))
51		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷))
52	50, 51	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐷 → ((𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ↔ (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷)))
53	48, 52	rspc 3276	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷)))
54	14, 43, 53	sylc 63	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷))
55		ptcnplem.6	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))
56	54, 55	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))
57	34	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ∈ 𝑉)
58		mptelixpg 7831	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ (𝐺‘𝑘)))
59	57, 58	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ((𝑘 ∈ 𝐼 ↦ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ (𝐺‘𝑘)))
60	56, 59	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ 𝐼 ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ (𝐺‘𝑘))
61	60	r19.21bi 2916	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ (𝐺‘𝑘))
62		cnpimaex 20870	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷) ∧ (𝐺‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ (𝐺‘𝑘)) → ∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘)))
63	11, 12, 61, 62	syl3anc 1318	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ 𝐼) → ∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘)))
64	9, 63	sylan2 490	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) → ∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘)))
65	64	ex 449	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ (𝐼 ∩ 𝑊) → ∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘))))
66	7, 65	ralrimi 2940	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ (𝐼 ∩ 𝑊)∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘)))
67		eleq2 2677	. . . . 5 ⊢ (𝑢 = (𝑓‘𝑘) → (𝐷 ∈ 𝑢 ↔ 𝐷 ∈ (𝑓‘𝑘)))
68		imaeq2 5381	. . . . . 6 ⊢ (𝑢 = (𝑓‘𝑘) → ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) = ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)))
69	68	sseq1d 3595	. . . . 5 ⊢ (𝑢 = (𝑓‘𝑘) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘) ↔ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))
70	67, 69	anbi12d 743	. . . 4 ⊢ (𝑢 = (𝑓‘𝑘) → ((𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘)) ↔ (𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘))))
71	70	ac6sfi 8089	. . 3 ⊢ (((𝐼 ∩ 𝑊) ∈ Fin ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)∃𝑢 ∈ 𝐽 (𝐷 ∈ 𝑢 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑢) ⊆ (𝐺‘𝑘))) → ∃𝑓(𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘))))
72	4, 66, 71	syl2anc 691	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑓(𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘))))
73	16	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝐽 ∈ (TopOn‘𝑋))
74		toponuni 20542	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
75	73, 74	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝑋 = ∪ 𝐽)
76	75	ineq1d 3775	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (𝑋 ∩ ∩ ran 𝑓) = (∪ 𝐽 ∩ ∩ ran 𝑓))
77		topontop 20541	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
78	16, 77	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
79	78	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝐽 ∈ Top)
80		frn 5966	. . . . . 6 ⊢ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 → ran 𝑓 ⊆ 𝐽)
81	80	ad2antrl 760	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ran 𝑓 ⊆ 𝐽)
82	4	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (𝐼 ∩ 𝑊) ∈ Fin)
83		ffn 5958	. . . . . . . 8 ⊢ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 → 𝑓 Fn (𝐼 ∩ 𝑊))
84	83	ad2antrl 760	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝑓 Fn (𝐼 ∩ 𝑊))
85		dffn4 6034	. . . . . . 7 ⊢ (𝑓 Fn (𝐼 ∩ 𝑊) ↔ 𝑓:(𝐼 ∩ 𝑊)–onto→ran 𝑓)
86	84, 85	sylib 207	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝑓:(𝐼 ∩ 𝑊)–onto→ran 𝑓)
87		fofi 8135	. . . . . 6 ⊢ (((𝐼 ∩ 𝑊) ∈ Fin ∧ 𝑓:(𝐼 ∩ 𝑊)–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
88	82, 86, 87	syl2anc 691	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ran 𝑓 ∈ Fin)
89		eqid 2610	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
90	89	rintopn 20539	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ ran 𝑓 ⊆ 𝐽 ∧ ran 𝑓 ∈ Fin) → (∪ 𝐽 ∩ ∩ ran 𝑓) ∈ 𝐽)
91	79, 81, 88, 90	syl3anc 1318	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (∪ 𝐽 ∩ ∩ ran 𝑓) ∈ 𝐽)
92	76, 91	eqeltrd 2688	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (𝑋 ∩ ∩ ran 𝑓) ∈ 𝐽)
93	13	ad2antrr 758	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝐷 ∈ 𝑋)
94		simpl 472	. . . . . . 7 ⊢ ((𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)) → 𝐷 ∈ (𝑓‘𝑘))
95	94	ralimi 2936	. . . . . 6 ⊢ (∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)) → ∀𝑘 ∈ (𝐼 ∩ 𝑊)𝐷 ∈ (𝑓‘𝑘))
96	95	ad2antll 761	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑘 ∈ (𝐼 ∩ 𝑊)𝐷 ∈ (𝑓‘𝑘))
97		eleq2 2677	. . . . . . 7 ⊢ (𝑧 = (𝑓‘𝑘) → (𝐷 ∈ 𝑧 ↔ 𝐷 ∈ (𝑓‘𝑘)))
98	97	ralrn 6270	. . . . . 6 ⊢ (𝑓 Fn (𝐼 ∩ 𝑊) → (∀𝑧 ∈ ran 𝑓 𝐷 ∈ 𝑧 ↔ ∀𝑘 ∈ (𝐼 ∩ 𝑊)𝐷 ∈ (𝑓‘𝑘)))
99	84, 98	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (∀𝑧 ∈ ran 𝑓 𝐷 ∈ 𝑧 ↔ ∀𝑘 ∈ (𝐼 ∩ 𝑊)𝐷 ∈ (𝑓‘𝑘)))
100	96, 99	mpbird 246	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑧 ∈ ran 𝑓 𝐷 ∈ 𝑧)
101		elrint 4453	. . . 4 ⊢ (𝐷 ∈ (𝑋 ∩ ∩ ran 𝑓) ↔ (𝐷 ∈ 𝑋 ∧ ∀𝑧 ∈ ran 𝑓 𝐷 ∈ 𝑧))
102	93, 100, 101	sylanbrc 695	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝐷 ∈ (𝑋 ∩ ∩ ran 𝑓))
103		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑓:(𝐼 ∩ 𝑊)⟶𝐽
104	7, 103	nfan 1816	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽)
105		funmpt 5840	. . . . . . . . . . . . 13 ⊢ Fun (𝑥 ∈ 𝑋 ↦ 𝐴)
106		simp-4l 802	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝜑)
107	106, 16	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝐽 ∈ (TopOn‘𝑋))
108		simpllr 795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝑓:(𝐼 ∩ 𝑊)⟶𝐽)
109		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝑘 ∈ (𝐼 ∩ 𝑊))
110	108, 109	ffvelrnd 6268	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (𝑓‘𝑘) ∈ 𝐽)
111		toponss 20544	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑓‘𝑘) ∈ 𝐽) → (𝑓‘𝑘) ⊆ 𝑋)
112	107, 110, 111	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (𝑓‘𝑘) ⊆ 𝑋)
113	8, 109	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝑘 ∈ 𝐼)
114	106, 113, 27	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘))
115		dmmptg 5549	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋)
116	114, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋)
117	112, 116	sseqtr4d 3605	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (𝑓‘𝑘) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐴))
118		funimass4 6157	. . . . . . . . . . . . 13 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ 𝐴) ∧ (𝑓‘𝑘) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘) ↔ ∀𝑡 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ (𝐺‘𝑘)))
119	105, 117, 118	sylancr 694	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘) ↔ ∀𝑡 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ (𝐺‘𝑘)))
120		nffvmpt1 6111	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡)
121	120	nfel1 2765	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ (𝐺‘𝑘)
122		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)
123		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥))
124	123	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ (𝐺‘𝑘) ↔ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)))
125	121, 122, 124	cbvral 3143	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘))
126	119, 125	syl6bb 275	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)))
127		inss1 3795	. . . . . . . . . . . . 13 ⊢ (𝑋 ∩ ∩ ran 𝑓) ⊆ 𝑋
128		ssralv 3629	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∩ ∩ ran 𝑓) ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘)))
129	127, 114, 128	mpsyl 66	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘))
130		inss2 3796	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∩ ∩ ran 𝑓) ⊆ ∩ ran 𝑓
131	108, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → 𝑓 Fn (𝐼 ∩ 𝑊))
132		fnfvelrn 6264	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 Fn (𝐼 ∩ 𝑊) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) → (𝑓‘𝑘) ∈ ran 𝑓)
133	131, 109, 132	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (𝑓‘𝑘) ∈ ran 𝑓)
134		intss1 4427	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑘) ∈ ran 𝑓 → ∩ ran 𝑓 ⊆ (𝑓‘𝑘))
135	133, 134	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → ∩ ran 𝑓 ⊆ (𝑓‘𝑘))
136	130, 135	syl5ss 3579	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (𝑋 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑘))
137		ssralv 3629	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∩ ∩ ran 𝑓) ⊆ (𝑓‘𝑘) → (∀𝑥 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)))
138	136, 137	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (∀𝑥 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)))
139		r19.26 3046	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)(𝐴 ∈ ∪ (𝐹‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)) ↔ (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘) ∧ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)))
140	127	sseli 3564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓) → 𝑥 ∈ 𝑋)
141	140, 29	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓) ∧ 𝐴 ∈ ∪ (𝐹‘𝑘)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
142	141	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓) ∧ 𝐴 ∈ ∪ (𝐹‘𝑘)) → (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘) ↔ 𝐴 ∈ (𝐺‘𝑘)))
143	142	biimpd 218	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓) ∧ 𝐴 ∈ ∪ (𝐹‘𝑘)) → (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘) → 𝐴 ∈ (𝐺‘𝑘)))
144	143	expimpd 627	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓) → ((𝐴 ∈ ∪ (𝐹‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)) → 𝐴 ∈ (𝐺‘𝑘)))
145	144	ralimia 2934	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)(𝐴 ∈ ∪ (𝐹‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
146	139, 145	sylbir 224	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘) ∧ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
147	129, 138, 146	syl6an 566	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (∀𝑥 ∈ (𝑓‘𝑘)((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) ∈ (𝐺‘𝑘) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
148	126, 147	sylbid 229	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) ∧ 𝐷 ∈ (𝑓‘𝑘)) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
149	148	expimpd 627	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼 ∩ 𝑊)) → ((𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
150	104, 149	ralimdaa 2941	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑓:(𝐼 ∩ 𝑊)⟶𝐽) → (∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)) → ∀𝑘 ∈ (𝐼 ∩ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
151	150	impr 647	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑘 ∈ (𝐼 ∩ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
152		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
153		eldifi 3694	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝐼 ∖ 𝑊) → 𝑘 ∈ 𝐼)
154	140, 28	sylan2 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)) → 𝐴 ∈ ∪ (𝐹‘𝑘))
155	154	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘))
156	152, 153, 155	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘))
157		ptcnplem.5	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → (𝐺‘𝑘) = ∪ (𝐹‘𝑘))
158		eleq2 2677	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑘) = ∪ (𝐹‘𝑘) → (𝐴 ∈ (𝐺‘𝑘) ↔ 𝐴 ∈ ∪ (𝐹‘𝑘)))
159	158	ralbidv 2969	. . . . . . . . . . . 12 ⊢ ((𝐺‘𝑘) = ∪ (𝐹‘𝑘) → (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘)))
160	157, 159	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ ∪ (𝐹‘𝑘)))
161	156, 160	mpbird 246	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ (𝐼 ∖ 𝑊)) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
162	161	ex 449	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑘 ∈ (𝐼 ∖ 𝑊) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
163	7, 162	ralrimi 2940	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑘 ∈ (𝐼 ∖ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
164	163	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑘 ∈ (𝐼 ∖ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
165		inundif 3998	. . . . . . . . 9 ⊢ ((𝐼 ∩ 𝑊) ∪ (𝐼 ∖ 𝑊)) = 𝐼
166	165	raleqi 3119	. . . . . . . 8 ⊢ (∀𝑘 ∈ ((𝐼 ∩ 𝑊) ∪ (𝐼 ∖ 𝑊))∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
167		ralunb 3756	. . . . . . . 8 ⊢ (∀𝑘 ∈ ((𝐼 ∩ 𝑊) ∪ (𝐼 ∖ 𝑊))∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ↔ (∀𝑘 ∈ (𝐼 ∩ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ∧ ∀𝑘 ∈ (𝐼 ∖ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
168	166, 167	bitr3i 265	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐼 ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ↔ (∀𝑘 ∈ (𝐼 ∩ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘) ∧ ∀𝑘 ∈ (𝐼 ∖ 𝑊)∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘)))
169	151, 164, 168	sylanbrc 695	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑘 ∈ 𝐼 ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
170		ralcom 3079	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)𝐴 ∈ (𝐺‘𝑘))
171	169, 170	sylibr 223	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘))
172	34	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → 𝐼 ∈ 𝑉)
173		nffvmpt1 6111	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡)
174	173	nfel1 2765	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)
175		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)
176		fveq2 6103	. . . . . . . . 9 ⊢ (𝑡 = 𝑥 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥))
177	176	eleq1d 2672	. . . . . . . 8 ⊢ (𝑡 = 𝑥 → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
178	174, 175, 177	cbvral 3143	. . . . . . 7 ⊢ (∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))
179	140, 36, 39	syl2anr 494	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) = (𝑘 ∈ 𝐼 ↦ 𝐴))
180	179	eleq1d 2672	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
181		mptelixpg 7831	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
182	181	adantr 480	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
183	180, 182	bitrd 267	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
184	183	ralbidva 2968	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑥) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
185	178, 184	syl5bb 271	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
186	172, 185	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑥 ∈ (𝑋 ∩ ∩ ran 𝑓)∀𝑘 ∈ 𝐼 𝐴 ∈ (𝐺‘𝑘)))
187	171, 186	mpbird 246	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘))
188		funmpt 5840	. . . . 5 ⊢ Fun (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))
189	34, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V)
190	189	ralrimivw 2950	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V)
191	190	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V)
192		dmmptg 5549	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ V → dom (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) = 𝑋)
193	191, 192	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → dom (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) = 𝑋)
194	127, 193	syl5sseqr 3617	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (𝑋 ∩ ∩ ran 𝑓) ⊆ dom (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)))
195		funimass4 6157	. . . . 5 ⊢ ((Fun (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∧ (𝑋 ∩ ∩ ran 𝑓) ⊆ dom (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
196	188, 194, 195	sylancr 694	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ∀𝑡 ∈ (𝑋 ∩ ∩ ran 𝑓)((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝑡) ∈ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
197	187, 196	mpbird 246	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘))
198		eleq2 2677	. . . . 5 ⊢ (𝑧 = (𝑋 ∩ ∩ ran 𝑓) → (𝐷 ∈ 𝑧 ↔ 𝐷 ∈ (𝑋 ∩ ∩ ran 𝑓)))
199		imaeq2 5381	. . . . . 6 ⊢ (𝑧 = (𝑋 ∩ ∩ ran 𝑓) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) = ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)))
200	199	sseq1d 3595	. . . . 5 ⊢ (𝑧 = (𝑋 ∩ ∩ ran 𝑓) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
201	198, 200	anbi12d 743	. . . 4 ⊢ (𝑧 = (𝑋 ∩ ∩ ran 𝑓) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)) ↔ (𝐷 ∈ (𝑋 ∩ ∩ ran 𝑓) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘))))
202	201	rspcev 3282	. . 3 ⊢ (((𝑋 ∩ ∩ ran 𝑓) ∈ 𝐽 ∧ (𝐷 ∈ (𝑋 ∩ ∩ ran 𝑓) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ (𝑋 ∩ ∩ ran 𝑓)) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
203	92, 102, 197, 202	syl12anc 1316	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑓:(𝐼 ∩ 𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼 ∩ 𝑊)(𝐷 ∈ (𝑓‘𝑘) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ (𝑓‘𝑘)) ⊆ (𝐺‘𝑘)))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))
204	72, 203	exlimddv 1850	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝐺‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 ∩ cint 4410 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 Fincfn 7841 ∏_tcpt 15922 Topctop 20517 TopOnctopon 20518 CnP ccnp 20839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-fin 7845 df-top 20521 df-topon 20523 df-cnp 20842

This theorem is referenced by: ptcnp 21235

W3C validator