Theorem bnj1463 31123

Description: Technical lemma for bnj60 31130. 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
bnj1463.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1463.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1463.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1463.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1463.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1463.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1463.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1463.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1463.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1463.10	⊢ 𝑃 = ∪ 𝐻
bnj1463.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1463.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1463.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1463.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1463.15	⊢ (𝜒 → 𝑄 ∈ V)
bnj1463.16	⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))
bnj1463.17	⊢ (𝜒 → 𝑄 Fn 𝐸)
bnj1463.18	⊢ (𝜒 → 𝐸 ∈ 𝐵)

Assertion

Ref	Expression
bnj1463	⊢ (𝜒 → 𝑄 ∈ 𝐶)

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐸,𝑑,𝑧   𝐺,𝑑,𝑓,𝑥,𝑧   𝑧,𝑄   𝑅,𝑑,𝑓,𝑥   𝑧,𝑌   𝑦,𝑑,𝑥

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦,𝑧)   𝐸(𝑥,𝑦,𝑓)   𝐺(𝑦)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1463

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1463.18	. . . . . . 7 ⊢ (𝜒 → 𝐸 ∈ 𝐵)
2	1	elexd 3214	. . . . . 6 ⊢ (𝜒 → 𝐸 ∈ V)
3		eleq1 2689	. . . . . . . 8 ⊢ (𝑑 = 𝐸 → (𝑑 ∈ 𝐵 ↔ 𝐸 ∈ 𝐵))
4		fneq2 5980	. . . . . . . . 9 ⊢ (𝑑 = 𝐸 → (𝑄 Fn 𝑑 ↔ 𝑄 Fn 𝐸))
5		raleq 3138	. . . . . . . . 9 ⊢ (𝑑 = 𝐸 → (∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊) ↔ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊)))
6	4, 5	anbi12d 747	. . . . . . . 8 ⊢ (𝑑 = 𝐸 → ((𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)) ↔ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))))
7	3, 6	anbi12d 747	. . . . . . 7 ⊢ (𝑑 = 𝐸 → ((𝑑 ∈ 𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))) ↔ (𝐸 ∈ 𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊)))))
8		bnj1463.1	. . . . . . . . . . . 12 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
9	8	bnj1317 30892	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 → ∀𝑑 𝑤 ∈ 𝐵)
10	9	nfcii 2755	. . . . . . . . . 10 ⊢ Ⅎ𝑑𝐵
11	10	nfel2 2781	. . . . . . . . 9 ⊢ Ⅎ𝑑 𝐸 ∈ 𝐵
12		bnj1463.2	. . . . . . . . . . . . 13 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13		bnj1463.3	. . . . . . . . . . . . 13 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
14		bnj1463.4	. . . . . . . . . . . . 13 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
15		bnj1463.5	. . . . . . . . . . . . 13 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
16		bnj1463.6	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
17		bnj1463.7	. . . . . . . . . . . . 13 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
18		bnj1463.8	. . . . . . . . . . . . 13 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
19		bnj1463.9	. . . . . . . . . . . . 13 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
20		bnj1463.10	. . . . . . . . . . . . 13 ⊢ 𝑃 = ∪ 𝐻
21		bnj1463.11	. . . . . . . . . . . . 13 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22		bnj1463.12	. . . . . . . . . . . . 13 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
23	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	bnj1467 31122	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄)
24	23	nfcii 2755	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝑄
25		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝐸
26	24, 25	nffn 5987	. . . . . . . . . 10 ⊢ Ⅎ𝑑 𝑄 Fn 𝐸
27		bnj1463.13	. . . . . . . . . . . . 13 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
28	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27	bnj1446 31113	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))
29	28	nf5i 2024	. . . . . . . . . . 11 ⊢ Ⅎ𝑑(𝑄‘𝑧) = (𝐺‘𝑊)
30	25, 29	nfral 2945	. . . . . . . . . 10 ⊢ Ⅎ𝑑∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊)
31	26, 30	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑑(𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))
32	11, 31	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑑(𝐸 ∈ 𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊)))
33	32	nf5ri 2065	. . . . . . 7 ⊢ ((𝐸 ∈ 𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))) → ∀𝑑(𝐸 ∈ 𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))))
34		bnj1463.17	. . . . . . . 8 ⊢ (𝜒 → 𝑄 Fn 𝐸)
35		bnj1463.16	. . . . . . . 8 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))
36	1, 34, 35	jca32 558	. . . . . . 7 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧 ∈ 𝐸 (𝑄‘𝑧) = (𝐺‘𝑊))))
37	7, 33, 36	bnj1465 30915	. . . . . 6 ⊢ ((𝜒 ∧ 𝐸 ∈ V) → ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
38	2, 37	mpdan 702	. . . . 5 ⊢ (𝜒 → ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
39		df-rex 2918	. . . . 5 ⊢ (∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)) ↔ ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
40	38, 39	sylibr 224	. . . 4 ⊢ (𝜒 → ∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)))
41		bnj1463.15	. . . . 5 ⊢ (𝜒 → 𝑄 ∈ V)
42		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑓𝐵
43	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	bnj1466 31121	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)
44	43	nfcii 2755	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝑄
45		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝑑
46	44, 45	nffn 5987	. . . . . . . . 9 ⊢ Ⅎ𝑓 𝑄 Fn 𝑑
47	8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27	bnj1448 31115	. . . . . . . . . . 11 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))
48	47	nf5i 2024	. . . . . . . . . 10 ⊢ Ⅎ𝑓(𝑄‘𝑧) = (𝐺‘𝑊)
49	45, 48	nfral 2945	. . . . . . . . 9 ⊢ Ⅎ𝑓∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)
50	46, 49	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))
51	42, 50	nfrex 3007	. . . . . . 7 ⊢ Ⅎ𝑓∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))
52	51	nf5ri 2065	. . . . . 6 ⊢ (∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)) → ∀𝑓∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)))
53	24	nfeq2 2780	. . . . . . 7 ⊢ Ⅎ𝑑 𝑓 = 𝑄
54		fneq1 5979	. . . . . . . 8 ⊢ (𝑓 = 𝑄 → (𝑓 Fn 𝑑 ↔ 𝑄 Fn 𝑑))
55		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑓 = 𝑄 → (𝑓‘𝑧) = (𝑄‘𝑧))
56		reseq1 5390	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑄 → (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)))
57	56	opeq2d 4409	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
58	57, 27	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = 𝑊)
59	58	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑓 = 𝑄 → (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) = (𝐺‘𝑊))
60	55, 59	eqeq12d 2637	. . . . . . . . 9 ⊢ (𝑓 = 𝑄 → ((𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ (𝑄‘𝑧) = (𝐺‘𝑊)))
61	60	ralbidv 2986	. . . . . . . 8 ⊢ (𝑓 = 𝑄 → (∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊)))
62	54, 61	anbi12d 747	. . . . . . 7 ⊢ (𝑓 = 𝑄 → ((𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
63	53, 62	rexbid 3051	. . . . . 6 ⊢ (𝑓 = 𝑄 → (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
64	52, 63, 43	bnj1468 30916	. . . . 5 ⊢ (𝑄 ∈ V → ([𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
65	41, 64	syl 17	. . . 4 ⊢ (𝜒 → ([𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑 ∈ 𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑄‘𝑧) = (𝐺‘𝑊))))
66	40, 65	mpbird 247	. . 3 ⊢ (𝜒 → [𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
67		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
68		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
69		bnj602 30985	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
70	69	reseq2d 5396	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
71	68, 70	opeq12d 4410	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
72	12, 71	syl5eq 2668	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑌 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
73	72	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑌) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
74	67, 73	eqeq12d 2637	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑓‘𝑥) = (𝐺‘𝑌) ↔ (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
75	74	cbvralv 3171	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
76	75	anbi2i 730	. . . . 5 ⊢ ((𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
77	76	rexbii 3041	. . . 4 ⊢ (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
78	77	sbcbii 3491	. . 3 ⊢ ([𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ [𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧 ∈ 𝑑 (𝑓‘𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
79	66, 78	sylibr 224	. 2 ⊢ (𝜒 → [𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
80	13	bnj1454 30912	. . 3 ⊢ (𝑄 ∈ V → (𝑄 ∈ 𝐶 ↔ [𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
81	41, 80	syl 17	. 2 ⊢ (𝜒 → (𝑄 ∈ 𝐶 ↔ [𝑄 / 𝑓]∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
82	79, 81	mpbird 247	1 ⊢ (𝜒 → 𝑄 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  [wsbc 3435   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114   ↾ cres 5116   Fn wfn 5883  ‘cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj14 30755

This theorem is referenced by:  bnj1312  31126