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Theorem bnj66 30638
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
bnj66.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj66.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj66.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
Assertion
Ref Expression
bnj66 (𝑔𝐶 → Rel 𝑔)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,𝑔   𝑓,𝐺,𝑔   𝑅,𝑓   𝑔,𝑌   𝑓,𝑑,𝑔   𝑥,𝑓,𝑔
Allowed substitution hints:   𝐴(𝑥,𝑔,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,𝑑)   𝑅(𝑥,𝑔,𝑑)   𝐺(𝑥,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj66
StepHypRef Expression
1 bnj66.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2 fneq1 5937 . . . . . . 7 (𝑔 = 𝑓 → (𝑔 Fn 𝑑𝑓 Fn 𝑑))
3 fveq1 6147 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑥) = (𝑓𝑥))
4 reseq1 5350 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
54opeq2d 4377 . . . . . . . . . . 11 (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
6 bnj66.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
75, 6syl6eqr 2673 . . . . . . . . . 10 (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
87fveq2d 6152 . . . . . . . . 9 (𝑔 = 𝑓 → (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
93, 8eqeq12d 2636 . . . . . . . 8 (𝑔 = 𝑓 → ((𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝑓𝑥) = (𝐺𝑌)))
109ralbidv 2980 . . . . . . 7 (𝑔 = 𝑓 → (∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
112, 10anbi12d 746 . . . . . 6 (𝑔 = 𝑓 → ((𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
1211rexbidv 3045 . . . . 5 (𝑔 = 𝑓 → (∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
1312cbvabv 2744 . . . 4 {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
141, 13eqtr4i 2646 . . 3 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
1514bnj1436 30618 . 2 (𝑔𝐶 → ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
16 bnj1239 30584 . 2 (∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → ∃𝑑𝐵 𝑔 Fn 𝑑)
17 fnrel 5947 . . 3 (𝑔 Fn 𝑑 → Rel 𝑔)
1817rexlimivw 3022 . 2 (∃𝑑𝐵 𝑔 Fn 𝑑 → Rel 𝑔)
1915, 16, 183syl 18 1 (𝑔𝐶 → Rel 𝑔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  wss 3555  cop 4154  cres 5076  Rel wrel 5079   Fn wfn 5842  cfv 5847   predc-bnj14 30461
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
This theorem is referenced by:  bnj1321  30803
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