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Theorem bnj1296 30797
Description: Technical lemma for bnj60 30838. 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
bnj1296.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1296.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1296.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1296.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1296.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1296.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
bnj1296.18 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
bnj1296.9 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.10 𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
bnj1296.11 𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.12 𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}
Assertion
Ref Expression
bnj1296 (𝜓 → (𝑔𝑥) = (𝑥))
Distinct variable groups:   𝐵,𝑓,𝑔   𝐵,,𝑓   𝑥,𝐷   𝐺,𝑑,𝑓,𝑔   ,𝐺,𝑑   𝑊,𝑑,𝑓   𝑔,𝑌   ,𝑌   𝑍,𝑑,𝑓   𝑥,𝑑,𝑓,𝑔   𝑥,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦)   𝐾(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐿(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑊(𝑥,𝑦,𝑔,)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑔,)

Proof of Theorem bnj1296
StepHypRef Expression
1 bnj1296.18 . . . . 5 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
21opeq2d 4377 . . . 4 (𝜓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3 bnj1296.9 . . . 4 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4 bnj1296.11 . . . 4 𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
52, 3, 43eqtr4g 2680 . . 3 (𝜓𝑍 = 𝑊)
65fveq2d 6152 . 2 (𝜓 → (𝐺𝑍) = (𝐺𝑊))
7 bnj1296.7 . . . 4 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
8 bnj1296.6 . . . . 5 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
9 bnj1296.10 . . . . . . . . . . 11 𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
109bnj1436 30618 . . . . . . . . . 10 (𝑔𝐾 → ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
11 fndm 5948 . . . . . . . . . . 11 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1211anim1i 591 . . . . . . . . . 10 ((𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)) → (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1310, 12bnj31 30493 . . . . . . . . 9 (𝑔𝐾 → ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
14 raleq 3127 . . . . . . . . . . 11 (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍) ↔ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1514pm5.32i 668 . . . . . . . . . 10 ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) ↔ (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1615rexbii 3034 . . . . . . . . 9 (∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) ↔ ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1713, 16sylibr 224 . . . . . . . 8 (𝑔𝐾 → ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)))
18 simpr 477 . . . . . . . 8 ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
1917, 18bnj31 30493 . . . . . . 7 (𝑔𝐾 → ∃𝑑𝐵𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
2019bnj1265 30591 . . . . . 6 (𝑔𝐾 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
21 bnj1296.2 . . . . . . 7 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22 bnj1296.3 . . . . . . 7 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2321, 22, 3, 9bnj1234 30789 . . . . . 6 𝐶 = 𝐾
2420, 23eleq2s 2716 . . . . 5 (𝑔𝐶 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
258, 24bnj770 30541 . . . 4 (𝜑 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
267, 25bnj835 30537 . . 3 (𝜓 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
27 bnj1296.4 . . . . 5 𝐷 = (dom 𝑔 ∩ dom )
2827bnj1292 30594 . . . 4 𝐷 ⊆ dom 𝑔
29 bnj1296.5 . . . . 5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029, 7bnj1212 30578 . . . 4 (𝜓𝑥𝐷)
3128, 30bnj1213 30577 . . 3 (𝜓𝑥 ∈ dom 𝑔)
3226, 31bnj1294 30596 . 2 (𝜓 → (𝑔𝑥) = (𝐺𝑍))
33 bnj1296.12 . . . . . . . . . . 11 𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}
3433bnj1436 30618 . . . . . . . . . 10 (𝐿 → ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
35 fndm 5948 . . . . . . . . . . 11 ( Fn 𝑑 → dom = 𝑑)
3635anim1i 591 . . . . . . . . . 10 (( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)) → (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
3734, 36bnj31 30493 . . . . . . . . 9 (𝐿 → ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
38 raleq 3127 . . . . . . . . . . 11 (dom = 𝑑 → (∀𝑥 ∈ dom (𝑥) = (𝐺𝑊) ↔ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
3938pm5.32i 668 . . . . . . . . . 10 ((dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) ↔ (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
4039rexbii 3034 . . . . . . . . 9 (∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) ↔ ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
4137, 40sylibr 224 . . . . . . . 8 (𝐿 → ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)))
42 simpr 477 . . . . . . . 8 ((dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4341, 42bnj31 30493 . . . . . . 7 (𝐿 → ∃𝑑𝐵𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4443bnj1265 30591 . . . . . 6 (𝐿 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4521, 22, 4, 33bnj1234 30789 . . . . . 6 𝐶 = 𝐿
4644, 45eleq2s 2716 . . . . 5 (𝐶 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
478, 46bnj771 30542 . . . 4 (𝜑 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
487, 47bnj835 30537 . . 3 (𝜓 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4927bnj1293 30595 . . . 4 𝐷 ⊆ dom
5049, 30bnj1213 30577 . . 3 (𝜓𝑥 ∈ dom )
5148, 50bnj1294 30596 . 2 (𝜓 → (𝑥) = (𝐺𝑊))
526, 32, 513eqtr4d 2665 1 (𝜓 → (𝑔𝑥) = (𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3554  wss 3555  cop 4154   class class class wbr 4613  dom cdm 5074  cres 5076   Fn wfn 5842  cfv 5847  w-bnj17 30459   predc-bnj14 30461   FrSe w-bnj15 30465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-bnj17 30460
This theorem is referenced by:  bnj1311  30800
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