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Theorem bnj1463 32435
Description: Technical lemma for bnj60 32442. 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 3464 . . . . . 6 (𝜒𝐸 ∈ V)
3 eleq1 2880 . . . . . . . 8 (𝑑 = 𝐸 → (𝑑𝐵𝐸𝐵))
4 fneq2 6419 . . . . . . . . 9 (𝑑 = 𝐸 → (𝑄 Fn 𝑑𝑄 Fn 𝐸))
5 raleq 3361 . . . . . . . . 9 (𝑑 = 𝐸 → (∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊) ↔ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))
64, 5anbi12d 633 . . . . . . . 8 (𝑑 = 𝐸 → ((𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) ↔ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))))
73, 6anbi12d 633 . . . . . . 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 32201 . . . . . . . . . . 11 (𝑤𝐵 → ∀𝑑 𝑤𝐵)
109nfcii 2943 . . . . . . . . . 10 𝑑𝐵
1110nfel2 2976 . . . . . . . . 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 32434 . . . . . . . . . . . 12 (𝑤𝑄 → ∀𝑑 𝑤𝑄)
2423nfcii 2943 . . . . . . . . . . 11 𝑑𝑄
25 nfcv 2958 . . . . . . . . . . 11 𝑑𝐸
2624, 25nffn 6426 . . . . . . . . . 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 32425 . . . . . . . . . . . 12 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))
2928nf5i 2148 . . . . . . . . . . 11 𝑑(𝑄𝑧) = (𝐺𝑊)
3025, 29nfralw 3192 . . . . . . . . . 10 𝑑𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)
3126, 30nfan 1900 . . . . . . . . 9 𝑑(𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
3211, 31nfan 1900 . . . . . . . 8 𝑑(𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊)))
3332nf5ri 2194 . . . . . . 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 519 . . . . . . 7 (𝜒 → (𝐸𝐵 ∧ (𝑄 Fn 𝐸 ∧ ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))))
377, 33, 36bnj1465 32225 . . . . . 6 ((𝜒𝐸 ∈ V) → ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
382, 37mpdan 686 . . . . 5 (𝜒 → ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
39 df-rex 3115 . . . . 5 (∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) ↔ ∃𝑑(𝑑𝐵 ∧ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
4038, 39sylibr 237 . . . 4 (𝜒 → ∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
41 bnj1463.15 . . . . 5 (𝜒𝑄 ∈ V)
42 nfcv 2958 . . . . . . . 8 𝑓𝐵
438, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22bnj1466 32433 . . . . . . . . . . 11 (𝑤𝑄 → ∀𝑓 𝑤𝑄)
4443nfcii 2943 . . . . . . . . . 10 𝑓𝑄
45 nfcv 2958 . . . . . . . . . 10 𝑓𝑑
4644, 45nffn 6426 . . . . . . . . 9 𝑓 𝑄 Fn 𝑑
478, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27bnj1448 32427 . . . . . . . . . . 11 ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))
4847nf5i 2148 . . . . . . . . . 10 𝑓(𝑄𝑧) = (𝐺𝑊)
4945, 48nfralw 3192 . . . . . . . . 9 𝑓𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)
5046, 49nfan 1900 . . . . . . . 8 𝑓(𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5142, 50nfrex 3271 . . . . . . 7 𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))
5251nf5ri 2194 . . . . . 6 (∃𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)) → ∀𝑓𝑑𝐵 (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
5324nfeq2 2975 . . . . . . 7 𝑑 𝑓 = 𝑄
54 fneq1 6418 . . . . . . . 8 (𝑓 = 𝑄 → (𝑓 Fn 𝑑𝑄 Fn 𝑑))
55 fveq1 6648 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝑓𝑧) = (𝑄𝑧))
56 reseq1 5816 . . . . . . . . . . . . 13 (𝑓 = 𝑄 → (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)))
5756opeq2d 4775 . . . . . . . . . . . 12 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
5857, 27eqtr4di 2854 . . . . . . . . . . 11 (𝑓 = 𝑄 → ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = 𝑊)
5958fveq2d 6653 . . . . . . . . . 10 (𝑓 = 𝑄 → (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) = (𝐺𝑊))
6055, 59eqeq12d 2817 . . . . . . . . 9 (𝑓 = 𝑄 → ((𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ (𝑄𝑧) = (𝐺𝑊)))
6160ralbidv 3165 . . . . . . . 8 (𝑓 = 𝑄 → (∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩) ↔ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊)))
6254, 61anbi12d 633 . . . . . . 7 (𝑓 = 𝑄 → ((𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)) ↔ (𝑄 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑄𝑧) = (𝐺𝑊))))
6353, 62rexbid 3282 . . . . . 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 32226 . . . . 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 260 . . 3 (𝜒[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
67 fveq2 6649 . . . . . . . 8 (𝑥 = 𝑧 → (𝑓𝑥) = (𝑓𝑧))
68 id 22 . . . . . . . . . . 11 (𝑥 = 𝑧𝑥 = 𝑧)
69 bnj602 32295 . . . . . . . . . . . 12 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
7069reseq2d 5822 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑧, 𝐴, 𝑅)))
7168, 70opeq12d 4776 . . . . . . . . . 10 (𝑥 = 𝑧 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7212, 71syl5eq 2848 . . . . . . . . 9 (𝑥 = 𝑧𝑌 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)
7372fveq2d 6653 . . . . . . . 8 (𝑥 = 𝑧 → (𝐺𝑌) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
7467, 73eqeq12d 2817 . . . . . . 7 (𝑥 = 𝑧 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7574cbvralvw 3399 . . . . . 6 (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩))
7675anbi2i 625 . . . . 5 ((𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7776rexbii 3213 . . . 4 (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑧𝑑 (𝑓𝑧) = (𝐺‘⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩)))
7877sbcbii 3779 . . 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 237 . 2 (𝜒[𝑄 / 𝑓]𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
8013bnj1454 32222 . . 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 260 1 (𝜒𝑄𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  [wsbc 3723  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  dom cdm 5523  cres 5525   Fn wfn 6323  cfv 6328   predc-bnj14 32066   FrSe w-bnj15 32070   trClc-bnj18 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-bnj14 32067
This theorem is referenced by:  bnj1312  32438
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