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Theorem bnj1463 31422
Description: Technical lemma for bnj60 31429. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1463.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1463.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1463.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1463.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1463.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1463.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
bnj1463.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1463.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1463.15 (𝜒𝑄 ∈ V)
bnj1463.16 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
bnj1463.17 (𝜒𝑄 Fn 𝐸)
bnj1463.18 (𝜒𝐸𝐵)
Assertion
Ref Expression
bnj1463 (𝜒𝑄𝐶)
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.
StepHypRef Expression
1 bnj1463.18 . . . . . . 7 (𝜒𝐸𝐵)
21elexd 3346 . . . . . 6 (𝜒𝐸 ∈ V)
3 eleq1 2819 . . . . . . . 8 (𝑑 = 𝐸 → (𝑑𝐵𝐸𝐵))
4 fneq2 6133 . . . . . . . . 9 (𝑑 = 𝐸 → (𝑄 Fn 𝑑𝑄 Fn 𝐸))
5 raleq 3269 . . . . . . . . 9 (𝑑 = 𝐸 → (∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊) ↔ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))
64, 5anbi12d 749 . . . . . . . 8 (𝑑 = 𝐸 → ((𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) ↔ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))))
73, 6anbi12d 749 . . . . . . 7 (𝑑 = 𝐸 → ((𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))) ↔ (𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))))
8 bnj1463.1 . . . . . . . . . . . 12 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1317 31191 . . . . . . . . . . 11 (𝑤𝐵 → ∀𝑑 𝑤𝐵)
109nfcii 2885 . . . . . . . . . 10 𝑑𝐵
1110nfel2 2911 . . . . . . . . 9 𝑑 𝐸𝐵
12 bnj1463.2 . . . . . . . . . . . . 13 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13 bnj1463.3 . . . . . . . . . . . . 13 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
14 bnj1463.4 . . . . . . . . . . . . 13 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
15 bnj1463.5 . . . . . . . . . . . . 13 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
16 bnj1463.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
17 bnj1463.7 . . . . . . . . . . . . 13 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
238, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22bnj1467 31421 . . . . . . . . . . . 12 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
2423nfcii 2885 . . . . . . . . . . 11 𝑑𝑄
25 nfcv 2894 . . . . . . . . . . 11 𝑑𝐸
2624, 25nffn 6140 . . . . . . . . . 10 𝑑 𝑄 Fn 𝐸
27 bnj1463.13 . . . . . . . . . . . . 13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
288, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27bnj1446 31412 . . . . . . . . . . . 12 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
2928nf5i 2165 . . . . . . . . . . 11 𝑑(𝑄𝑧) = (𝐺𝑊)
3025, 29nfral 3075 . . . . . . . . . 10 𝑑𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)
3126, 30nfan 1969 . . . . . . . . 9 𝑑(𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
3211, 31nfan 1969 . . . . . . . 8 𝑑(𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))
3332nf5ri 2204 . . . . . . 7 ((𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))) → ∀𝑑(𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))))
34 bnj1463.17 . . . . . . . 8 (𝜒𝑄 Fn 𝐸)
35 bnj1463.16 . . . . . . . 8 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
361, 34, 35jca32 559 . . . . . . 7 (𝜒 → (𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))))
377, 33, 36bnj1465 31214 . . . . . 6 ((𝜒𝐸 ∈ V) → ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
382, 37mpdan 705 . . . . 5 (𝜒 → ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
39 df-rex 3048 . . . . 5 (∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) ↔ ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
4038, 39sylibr 224 . . . 4 (𝜒 → ∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
41 bnj1463.15 . . . . 5 (𝜒𝑄 ∈ V)
42 nfcv 2894 . . . . . . . 8 𝑓𝐵
438, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22bnj1466 31420 . . . . . . . . . . 11 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
4443nfcii 2885 . . . . . . . . . 10 𝑓𝑄
45 nfcv 2894 . . . . . . . . . 10 𝑓𝑑
4644, 45nffn 6140 . . . . . . . . 9 𝑓 𝑄 Fn 𝑑
478, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27bnj1448 31414 . . . . . . . . . . 11 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
4847nf5i 2165 . . . . . . . . . 10 𝑓(𝑄𝑧) = (𝐺𝑊)
4945, 48nfral 3075 . . . . . . . . 9 𝑓𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)
5046, 49nfan 1969 . . . . . . . 8 𝑓(𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5142, 50nfrex 3137 . . . . . . 7 𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5251nf5ri 2204 . . . . . 6 (∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) → ∀𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
5324nfeq2 2910 . . . . . . 7 𝑑 𝑓 = 𝑄
54 fneq1 6132 . . . . . . . 8 (𝑓 = 𝑄 → (𝑓 Fn 𝑑𝑄 Fn 𝑑))
55 fveq1 6343 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝑓𝑧) = (𝑄𝑧))
56 reseq1 5537 . . . . . . . . . . . . 13 (𝑓 = 𝑄 → (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)))
5756opeq2d 4552 . . . . . . . . . . . 12 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
5857, 27syl6eqr 2804 . . . . . . . . . . 11 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = 𝑊)
5958fveq2d 6348 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) = (𝐺𝑊))
6055, 59eqeq12d 2767 . . . . . . . . 9 (𝑓 = 𝑄 → ((𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ (𝑄𝑧) = (𝐺𝑊)))
6160ralbidv 3116 . . . . . . . 8 (𝑓 = 𝑄 → (∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
6254, 61anbi12d 749 . . . . . . 7 (𝑓 = 𝑄 → ((𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
6353, 62rexbid 3181 . . . . . 6 (𝑓 = 𝑄 → (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
6452, 63, 43bnj1468 31215 . . . . 5 (𝑄 ∈ V → ([𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
6541, 64syl 17 . . . 4 (𝜒 → ([𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ ∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
6640, 65mpbird 247 . . 3 (𝜒[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
67 fveq2 6344 . . . . . . . 8 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
68 id 22 . . . . . . . . . . 11 (𝑥 = 𝑧𝑥 = 𝑧)
69 bnj602 31284 . . . . . . . . . . . 12 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
7069reseq2d 5543 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
7168, 70opeq12d 4553 . . . . . . . . . 10 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7212, 71syl5eq 2798 . . . . . . . . 9 (𝑥 = 𝑧𝑌 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7372fveq2d 6348 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
7467, 73eqeq12d 2767 . . . . . . 7 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7574cbvralv 3302 . . . . . 6 (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
7675anbi2i 732 . . . . 5 ((𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7776rexbii 3171 . . . 4 (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7877sbcbii 3624 . . 3 ([𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ [𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7966, 78sylibr 224 . 2 (𝜒[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
8013bnj1454 31211 . . 3 (𝑄 ∈ V → (𝑄𝐶[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
8141, 80syl 17 . 2 (𝜒 → (𝑄𝐶[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
8279, 81mpbird 247 1 (𝜒𝑄𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wne 2924  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  [wsbc 3568  cun 3705  wss 3707  c0 4050  {csn 4313  cop 4319   cuni 4580   class class class wbr 4796  dom cdm 5258  cres 5260   Fn wfn 6036  cfv 6041   predc-bnj14 31055   FrSe w-bnj15 31059   trClc-bnj18 31061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fn 6044  df-fv 6049  df-bnj14 31056
This theorem is referenced by:  bnj1312  31425
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