Theorem bnj1118 30306

Description: Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1118.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1118.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1118.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1118.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1118.18	⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
bnj1118.19	⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
bnj1118.26	⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Assertion

Ref	Expression
bnj1118	⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Distinct variable groups:   𝐴,𝑖,𝑗,𝑦   𝑦,𝐵   𝐷,𝑗   𝑅,𝑖,𝑗,𝑦   𝑓,𝑖,𝑗,𝑦   𝑖,𝑛,𝑗

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜏(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑓,𝑛)   𝐵(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑖,𝑛)   𝑅(𝑓,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜑₀(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1118

Step	Hyp	Ref	Expression
1		bnj1118.3	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj1118.7	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
3		bnj1118.18	. . . 4 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
4		bnj1118.19	. . . 4 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
5		bnj1118.26	. . . 4 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
6	1, 2, 3, 4, 5	bnj1110 30304	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))
7		ancl 567	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))))
8	6, 7	bnj101 30043	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)))
9		simpr2 1061	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 = suc 𝑗)
10	1	bnj1254 30134	. . . . . . 7 ⊢ (𝜒 → 𝜓)
11	10	3ad2ant3 1077	. . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝜓)
12	11	ad2antrl 760	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜓)
13	12	adantr 480	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝜓)
14	1	bnj1232 30128	. . . . . . . . 9 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
15	14	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
16	15	ad2antrl 760	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑛 ∈ 𝐷)
17	16	adantr 480	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑛 ∈ 𝐷)
18		simpr1 1060	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ 𝑛)
19	2	bnj923 30092	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
20	19	anim1i 590	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑛 ∈ ω ∧ 𝑗 ∈ 𝑛))
21	20	ancomd 466	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑗 ∈ 𝑛) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
22	17, 18, 21	syl2anc 691	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω))
23		elnn 6967	. . . . 5 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω)
24	22, 23	syl 17	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑗 ∈ ω)
25	4	bnj1232 30128	. . . . . 6 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
26	25	adantl 481	. . . . 5 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → 𝑖 ∈ 𝑛)
27	26	ad2antlr 759	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → 𝑖 ∈ 𝑛)
28	9, 13, 24, 27	bnj951 30100	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛))
29		bnj1118.5	. . . . . . 7 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
30	29	simp2bi 1070	. . . . . 6 ⊢ (𝜏 → TrFo(𝐵, 𝐴, 𝑅))
31	30	3ad2ant2 1076	. . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → TrFo(𝐵, 𝐴, 𝑅))
32	31	ad2antrl 760	. . . 4 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → TrFo(𝐵, 𝐴, 𝑅))
33		simp3 1056	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵) → (𝑓‘𝑗) ⊆ 𝐵)
34	32, 33	anim12i 588	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵))
35		bnj256 30025	. . . . 5 ⊢ ((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ↔ ((𝑖 = suc 𝑗 ∧ 𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛)))
36		bnj1118.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
37	36	bnj1112 30305	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
38	37	biimpi 205	. . . . . . . 8 ⊢ (𝜓 → ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
39	38	19.21bi 2047	. . . . . . 7 ⊢ (𝜓 → ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
40		eleq1 2676	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
41	40	anbi2d 736	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
42		fveq2 6103	. . . . . . . . 9 ⊢ (𝑖 = suc 𝑗 → (𝑓‘𝑖) = (𝑓‘suc 𝑗))
43	42	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
44	41, 43	imbi12d 333	. . . . . . 7 ⊢ (𝑖 = suc 𝑗 → (((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
45	39, 44	syl5ibr 235	. . . . . 6 ⊢ (𝑖 = suc 𝑗 → (𝜓 → ((𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
46	45	imp31 447	. . . . 5 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓) ∧ (𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
47	35, 46	sylbi 206	. . . 4 ⊢ ((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
48		df-bnj19 30016	. . . . . . 7 ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
49		ssralv 3629	. . . . . . 7 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
50	48, 49	syl5bi 231	. . . . . 6 ⊢ ((𝑓‘𝑗) ⊆ 𝐵 → ( TrFo(𝐵, 𝐴, 𝑅) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
51	50	impcom 445	. . . . 5 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
52		iunss 4497	. . . . 5 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵 ↔ ∀𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
53	51, 52	sylibr 223	. . . 4 ⊢ (( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵) → ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵)
54		sseq1 3589	. . . . 5 ⊢ ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) → ((𝑓‘𝑖) ⊆ 𝐵 ↔ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵))
55	54	biimpar 501	. . . 4 ⊢ (((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ∧ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵)
56	47, 53, 55	syl2an 493	. . 3 ⊢ (((𝑖 = suc 𝑗 ∧ 𝜓 ∧ 𝑗 ∈ ω ∧ 𝑖 ∈ 𝑛) ∧ ( TrFo(𝐵, 𝐴, 𝑅) ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
57	28, 34, 56	syl2anc 691	. 2 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵)) → (𝑓‘𝑖) ⊆ 𝐵)
58	8, 57	bnj1023 30105	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455   class class class wbr 4583   E cep 4947  dom cdm 5038  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   TrFow-bnj19 30015

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-sep 4709  ax-nul 4717  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-rab 2905  df-v 3175  df-sbc 3403  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-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fn 5807  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj19 30016

This theorem is referenced by:  bnj1030  30309