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Theorem bnj66 34718
Description: Technical lemma for bnj60 34920. 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 6643 . . . . . . 7 (𝑔 = 𝑓 → (𝑔 Fn 𝑑𝑓 Fn 𝑑))
3 fveq1 6892 . . . . . . . . 9 (𝑔 = 𝑓 → (𝑔𝑥) = (𝑓𝑥))
4 reseq1 5975 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
54opeq2d 4878 . . . . . . . . . . 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, 6eqtr4di 2784 . . . . . . . . . 10 (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
87fveq2d 6897 . . . . . . . . 9 (𝑔 = 𝑓 → (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
93, 8eqeq12d 2742 . . . . . . . 8 (𝑔 = 𝑓 → ((𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝑓𝑥) = (𝐺𝑌)))
109ralbidv 3168 . . . . . . 7 (𝑔 = 𝑓 → (∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
112, 10anbi12d 630 . . . . . 6 (𝑔 = 𝑓 → ((𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
1211rexbidv 3169 . . . . 5 (𝑔 = 𝑓 → (∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))))
1312cbvabv 2799 . . . 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 2757 . . 3 𝐶 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
1514bnj1436 34697 . 2 (𝑔𝐶 → ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
16 bnj1239 34663 . 2 (∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → ∃𝑑𝐵 𝑔 Fn 𝑑)
17 fnrel 6654 . . 3 (𝑔 Fn 𝑑 → Rel 𝑔)
1817rexlimivw 3141 . 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 394   = wceq 1534  wcel 2099  {cab 2703  wral 3051  wrex 3060  wss 3946  cop 4629  cres 5676  Rel wrel 5679   Fn wfn 6541  cfv 6546   predc-bnj14 34546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-res 5686  df-iota 6498  df-fun 6548  df-fn 6549  df-fv 6554
This theorem is referenced by:  bnj1321  34885
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