Theorem iundom2g 9241

Description: An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
iunfo.1	⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
iundomg.2	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∈ AC 𝐴)
iundomg.3	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)

Assertion

Ref	Expression
iundom2g	⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑇(𝑥)

Proof of Theorem iundom2g

Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iundomg.2	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∈ AC 𝐴)
2		iundomg.3	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)
3		brdomi 7852	. . . . . . . . 9 ⊢ (𝐵 ≼ 𝐶 → ∃𝑔 𝑔:𝐵–1-1→𝐶)
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔 𝑔:𝐵–1-1→𝐶)
5		f1f 6014	. . . . . . . . . . . 12 ⊢ (𝑔:𝐵–1-1→𝐶 → 𝑔:𝐵⟶𝐶)
6		reldom 7847	. . . . . . . . . . . . . . 15 ⊢ Rel ≼
7	6	brrelex2i 5083	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≼ 𝐶 → 𝐶 ∈ V)
8	6	brrelexi 5082	. . . . . . . . . . . . . 14 ⊢ (𝐵 ≼ 𝐶 → 𝐵 ∈ V)
9	7, 8	elmapd 7758	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ 𝐶 → (𝑔 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶𝐶))
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔 ∈ (𝐶 ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶𝐶))
11	5, 10	syl5ibr 235	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → 𝑔 ∈ (𝐶 ↑𝑚 𝐵)))
12		ssiun2 4499	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ↑𝑚 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵))
13	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝐶 ↑𝑚 𝐵) ⊆ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵))
14	13	sseld 3567	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔 ∈ (𝐶 ↑𝑚 𝐵) → 𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)))
15	11, 14	syld 46	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → 𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)))
16	15	ancrd 575	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (𝑔:𝐵–1-1→𝐶 → (𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ 𝑔:𝐵–1-1→𝐶)))
17	16	eximdv 1833	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → (∃𝑔 𝑔:𝐵–1-1→𝐶 → ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ 𝑔:𝐵–1-1→𝐶)))
18	4, 17	mpd 15	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ 𝑔:𝐵–1-1→𝐶))
19		df-rex 2902	. . . . . . 7 ⊢ (∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∃𝑔(𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ 𝑔:𝐵–1-1→𝐶))
20	18, 19	sylibr 223	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ≼ 𝐶) → ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶)
21	20	ralimiaa 2935	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → ∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶)
22	2, 21	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶)
23		nfv 1830	. . . . 5 ⊢ Ⅎ𝑦∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶
24		nfiu1 4486	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)
25		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑔
26		nfcsb1v 3515	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
27		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
28	25, 26, 27	nff1 6012	. . . . . 6 ⊢ Ⅎ𝑥 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
29	24, 28	nfrex 2990	. . . . 5 ⊢ Ⅎ𝑥∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
30		csbeq1a 3508	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
31		f1eq2 6010	. . . . . . 7 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 → (𝑔:𝐵–1-1→𝐶 ↔ 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
32	30, 31	syl 17	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑔:𝐵–1-1→𝐶 ↔ 𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
33	32	rexbidv 3034	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
34	23, 29, 33	cbvral 3143	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
35	22, 34	sylib 207	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
36		f1eq1 6009	. . . 4 ⊢ (𝑔 = (𝑓‘𝑦) → (𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
37	36	acni3 8753	. . 3 ⊢ ((∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∈ AC 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑔 ∈ ∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵)𝑔:⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → ∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
38	1, 35, 37	syl2anc 691	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
39		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦(𝑓‘𝑥):𝐵–1-1→𝐶
40		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑦)
41	40, 26, 27	nff1 6012	. . . . . 6 ⊢ Ⅎ𝑥(𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶
42		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
43		f1eq1 6009	. . . . . . . 8 ⊢ ((𝑓‘𝑥) = (𝑓‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):𝐵–1-1→𝐶))
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):𝐵–1-1→𝐶))
45		f1eq2 6010	. . . . . . . 8 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 → ((𝑓‘𝑦):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
46	30, 45	syl 17	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑦):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
47	44, 46	bitrd 267	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶))
48	39, 41, 47	cbvral 3143	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶)
49		df-ne 2782	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
50		acnrcl 8748	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∈ AC 𝐴 → 𝐴 ∈ V)
51	1, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
52		r19.2z 4012	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ≼ 𝐶)
53	7	rexlimivw 3011	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → 𝐶 ∈ V)
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) → 𝐶 ∈ V)
55	54	expcom 450	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶 → (𝐴 ≠ ∅ → 𝐶 ∈ V))
56	2, 55	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ≠ ∅ → 𝐶 ∈ V))
57		xpexg 6858	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
58	51, 56, 57	syl6an 566	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐴 × 𝐶) ∈ V))
59	49, 58	syl5bir 232	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = ∅ → (𝐴 × 𝐶) ∈ V))
60		xpeq1 5052	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) = (∅ × 𝐶))
61		0xp 5122	. . . . . . . . 9 ⊢ (∅ × 𝐶) = ∅
62		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
63	61, 62	eqeltri 2684	. . . . . . . 8 ⊢ (∅ × 𝐶) ∈ V
64	60, 63	syl6eqel 2696	. . . . . . 7 ⊢ (𝐴 = ∅ → (𝐴 × 𝐶) ∈ V)
65	59, 64	pm2.61d2 171	. . . . . 6 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V)
66		iunfo.1	. . . . . . . . . 10 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
67	66	eleq2i 2680	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
68		eliun 4460	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵))
69	67, 68	bitri 263	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑇 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵))
70		r19.29 3054	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → ∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)))
71		xp1st 7089	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑥} × 𝐵) → (1st ‘𝑦) ∈ {𝑥})
72	71	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) ∈ {𝑥})
73		elsni 4142	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑦) ∈ {𝑥} → (1st ‘𝑦) = 𝑥)
74	72, 73	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) = 𝑥)
75		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝑥 ∈ 𝐴)
76	74, 75	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (1st ‘𝑦) ∈ 𝐴)
77	74	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (𝑓‘(1st ‘𝑦)) = (𝑓‘𝑥))
78	77	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘𝑥)‘(2nd ‘𝑦)))
79		f1f 6014	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑓‘𝑥):𝐵⟶𝐶)
80	79	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (𝑓‘𝑥):𝐵⟶𝐶)
81		xp2nd 7090	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
82	81	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (2nd ‘𝑦) ∈ 𝐵)
83	80, 82	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘𝑥)‘(2nd ‘𝑦)) ∈ 𝐶)
84	78, 83	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) ∈ 𝐶)
85		opelxpi 5072	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑦) ∈ 𝐴 ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) ∈ 𝐶) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
86	76, 84, 85	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
87	86	rexlimiva 3010	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
88	70, 87	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶))
89	88	ex 449	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶)))
90	69, 89	syl5bi 231	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑦 ∈ 𝑇 → ⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ ∈ (𝐴 × 𝐶)))
91		fvex 6113	. . . . . . . . . 10 ⊢ (1st ‘𝑦) ∈ V
92		fvex 6113	. . . . . . . . . 10 ⊢ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) ∈ V
93	91, 92	opth 4871	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ ((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))))
94		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (1st ‘𝑦) = (1st ‘𝑧))
95	94	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (𝑓‘(1st ‘𝑦)) = (𝑓‘(1st ‘𝑧)))
96	95	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧)))
97	96	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))))
98		djussxp 5189	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
99	66, 98	eqsstri 3598	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ⊆ (𝐴 × V)
100		simprl 790	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑦 ∈ 𝑇)
101	99, 100	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑦 ∈ (𝐴 × V))
102	101	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → 𝑦 ∈ (𝐴 × V))
103		xp1st 7089	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝐴 × V) → (1st ‘𝑦) ∈ 𝐴)
104	102, 103	syl 17	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (1st ‘𝑦) ∈ 𝐴)
105		simpll 786	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶)
106		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘(1st ‘𝑦))
107		nfcsb1v 3515	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋(1st ‘𝑦) / 𝑥⦌𝐵
108	106, 107, 27	nff1 6012	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶
109		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑦) → (𝑓‘𝑥) = (𝑓‘(1st ‘𝑦)))
110		f1eq1 6009	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) = (𝑓‘(1st ‘𝑦)) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶))
111	109, 110	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶))
112		csbeq1a 3508	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑦) → 𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
113		f1eq2 6010	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵 → ((𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
114	112, 113	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘(1st ‘𝑦)):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
115	111, 114	bitrd 267	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑦) → ((𝑓‘𝑥):𝐵–1-1→𝐶 ↔ (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
116	108, 115	rspc 3276	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑦) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶))
117	104, 105, 116	sylc 63	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶)
118	107	nfel2 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵
119	74	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝑥 = (1st ‘𝑦))
120	119, 112	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → 𝐵 = ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
121	82, 120	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ∧ ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵))) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
122	121	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐴 → (((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
123	118, 122	rexlimi 3006	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥 ∈ 𝐴 ((𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
124	70, 123	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
125	124	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ ({𝑥} × 𝐵) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
126	69, 125	syl5bi 231	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → (𝑦 ∈ 𝑇 → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵))
127	126	imp 444	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑦 ∈ 𝑇) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
128	127	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
129	128	adantr 480	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
130	126	ralrimiv 2948	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ∀𝑦 ∈ 𝑇 (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
131		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (2nd ‘𝑦) = (2nd ‘𝑧))
132		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → (1st ‘𝑦) = (1st ‘𝑧))
133	132	csbeq1d 3506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ⦋(1st ‘𝑦) / 𝑥⦌𝐵 = ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
134	131, 133	eleq12d 2682	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ↔ (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵))
135	134	rspccva 3281	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑇 (2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ∧ 𝑧 ∈ 𝑇) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
136	130, 135	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ 𝑧 ∈ 𝑇) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
137	136	adantrl 748	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
138	137	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
139	94	csbeq1d 3506	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → ⦋(1st ‘𝑦) / 𝑥⦌𝐵 = ⦋(1st ‘𝑧) / 𝑥⦌𝐵)
140	138, 139	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (2nd ‘𝑧) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)
141		f1fveq 6420	. . . . . . . . . . . . 13 ⊢ (((𝑓‘(1st ‘𝑦)):⦋(1st ‘𝑦) / 𝑥⦌𝐵–1-1→𝐶 ∧ ((2nd ‘𝑦) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵 ∧ (2nd ‘𝑧) ∈ ⦋(1st ‘𝑦) / 𝑥⦌𝐵)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
142	117, 129, 140, 141	syl12anc 1316	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
143	97, 142	bitr3d 269	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) ∧ (1st ‘𝑦) = (1st ‘𝑧)) → (((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧)) ↔ (2nd ‘𝑦) = (2nd ‘𝑧)))
144	143	pm5.32da 671	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))) ↔ ((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧))))
145		simprr 792	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑧 ∈ 𝑇)
146	99, 145	sseldi 3566	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → 𝑧 ∈ (𝐴 × V))
147		xpopth 7098	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 × V) ∧ 𝑧 ∈ (𝐴 × V)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧)) ↔ 𝑦 = 𝑧))
148	101, 146, 147	syl2anc 691	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ (2nd ‘𝑦) = (2nd ‘𝑧)) ↔ 𝑦 = 𝑧))
149	144, 148	bitrd 267	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (((1st ‘𝑦) = (1st ‘𝑧) ∧ ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦)) = ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))) ↔ 𝑦 = 𝑧))
150	93, 149	syl5bb 271	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ 𝑦 = 𝑧))
151	150	ex 449	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (⟨(1st ‘𝑦), ((𝑓‘(1st ‘𝑦))‘(2nd ‘𝑦))⟩ = ⟨(1st ‘𝑧), ((𝑓‘(1st ‘𝑧))‘(2nd ‘𝑧))⟩ ↔ 𝑦 = 𝑧)))
152	90, 151	dom2d 7882	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → ((𝐴 × 𝐶) ∈ V → 𝑇 ≼ (𝐴 × 𝐶)))
153	65, 152	syl5com 31	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥):𝐵–1-1→𝐶 → 𝑇 ≼ (𝐴 × 𝐶)))
154	48, 153	syl5bir 232	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶 → 𝑇 ≼ (𝐴 × 𝐶)))
155	154	adantld 482	. . 3 ⊢ (𝜑 → ((𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → 𝑇 ≼ (𝐴 × 𝐶)))
156	155	exlimdv 1848	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝐴⟶∪ 𝑥 ∈ 𝐴 (𝐶 ↑𝑚 𝐵) ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦):⦋𝑦 / 𝑥⦌𝐵–1-1→𝐶) → 𝑇 ≼ (𝐴 × 𝐶)))
157	38, 156	mpd 15	1 ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ⦋csb 3499   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   × cxp 5036  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744   ≼ cdom 7839  AC wacn 8647

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-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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-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-1st 7059  df-2nd 7060  df-map 7746  df-dom 7843  df-acn 8651

This theorem is referenced by:  iundomg  9242  iundom  9243