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Theorem bnj1463 31351
 Description: Technical lemma for bnj60 31358. 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 3318 . . . . . 6 (𝜒𝐸 ∈ V)
3 eleq1 2791 . . . . . . . 8 (𝑑 = 𝐸 → (𝑑𝐵𝐸𝐵))
4 fneq2 6093 . . . . . . . . 9 (𝑑 = 𝐸 → (𝑄 Fn 𝑑𝑄 Fn 𝐸))
5 raleq 3241 . . . . . . . . 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 31120 . . . . . . . . . . 11 (𝑤𝐵 → ∀𝑑 𝑤𝐵)
109nfcii 2857 . . . . . . . . . 10 𝑑𝐵
1110nfel2 2883 . . . . . . . . 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 31350 . . . . . . . . . . . 12 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
2423nfcii 2857 . . . . . . . . . . 11 𝑑𝑄
25 nfcv 2866 . . . . . . . . . . 11 𝑑𝐸
2624, 25nffn 6100 . . . . . . . . . 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 31341 . . . . . . . . . . . 12 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
2928nf5i 2137 . . . . . . . . . . 11 𝑑(𝑄𝑧) = (𝐺𝑊)
3025, 29nfral 3047 . . . . . . . . . 10 𝑑𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)
3126, 30nfan 1941 . . . . . . . . 9 𝑑(𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
3211, 31nfan 1941 . . . . . . . 8 𝑑(𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))
3332nf5ri 2176 . . . . . . 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 31143 . . . . . 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 3020 . . . . 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 2866 . . . . . . . 8 𝑓𝐵
438, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22bnj1466 31349 . . . . . . . . . . 11 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
4443nfcii 2857 . . . . . . . . . 10 𝑓𝑄
45 nfcv 2866 . . . . . . . . . 10 𝑓𝑑
4644, 45nffn 6100 . . . . . . . . 9 𝑓 𝑄 Fn 𝑑
478, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27bnj1448 31343 . . . . . . . . . . 11 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
4847nf5i 2137 . . . . . . . . . 10 𝑓(𝑄𝑧) = (𝐺𝑊)
4945, 48nfral 3047 . . . . . . . . 9 𝑓𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)
5046, 49nfan 1941 . . . . . . . 8 𝑓(𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5142, 50nfrex 3109 . . . . . . 7 𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5251nf5ri 2176 . . . . . 6 (∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) → ∀𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
5324nfeq2 2882 . . . . . . 7 𝑑 𝑓 = 𝑄
54 fneq1 6092 . . . . . . . 8 (𝑓 = 𝑄 → (𝑓 Fn 𝑑𝑄 Fn 𝑑))
55 fveq1 6303 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝑓𝑧) = (𝑄𝑧))
56 reseq1 5497 . . . . . . . . . . . . 13 (𝑓 = 𝑄 → (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)))
5756opeq2d 4516 . . . . . . . . . . . 12 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
5857, 27syl6eqr 2776 . . . . . . . . . . 11 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = 𝑊)
5958fveq2d 6308 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) = (𝐺𝑊))
6055, 59eqeq12d 2739 . . . . . . . . 9 (𝑓 = 𝑄 → ((𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ (𝑄𝑧) = (𝐺𝑊)))
6160ralbidv 3088 . . . . . . . 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 3153 . . . . . 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 31144 . . . . 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 6304 . . . . . . . 8 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
68 id 22 . . . . . . . . . . 11 (𝑥 = 𝑧𝑥 = 𝑧)
69 bnj602 31213 . . . . . . . . . . . 12 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
7069reseq2d 5503 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
7168, 70opeq12d 4517 . . . . . . . . . 10 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7212, 71syl5eq 2770 . . . . . . . . 9 (𝑥 = 𝑧𝑌 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7372fveq2d 6308 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
7467, 73eqeq12d 2739 . . . . . . 7 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7574cbvralv 3274 . . . . . 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 3143 . . . 4 (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7877sbcbii 3597 . . 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 31140 . . 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 1596  ∃wex 1817   ∈ wcel 2103  {cab 2710   ≠ wne 2896  ∀wral 3014  ∃wrex 3015  {crab 3018  Vcvv 3304  [wsbc 3541   ∪ cun 3678   ⊆ wss 3680  ∅c0 4023  {csn 4285  ⟨cop 4291  ∪ cuni 4544   class class class wbr 4760  dom cdm 5218   ↾ cres 5220   Fn wfn 5996  ‘cfv 6001   predc-bnj14 30984   FrSe w-bnj15 30988   trClc-bnj18 30990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-res 5230  df-iota 5964  df-fun 6003  df-fn 6004  df-fv 6009  df-bnj14 30985 This theorem is referenced by:  bnj1312  31354
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