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Theorem bnj1234 32706
Description: Technical lemma for bnj60 32755. 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
bnj1234.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1234.4 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1234.5 𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
Assertion
Ref Expression
bnj1234 𝐶 = 𝐷
Distinct variable groups:   𝐵,𝑓,𝑔   𝑓,𝐺,𝑔   𝑔,𝑌   𝑓,𝑍   𝑓,𝑑,𝑔   𝑥,𝑓,𝑔
Allowed substitution hints:   𝐴(𝑥,𝑓,𝑔,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑔,𝑑)   𝐷(𝑥,𝑓,𝑔,𝑑)   𝑅(𝑥,𝑓,𝑔,𝑑)   𝐺(𝑥,𝑑)   𝑌(𝑥,𝑓,𝑑)   𝑍(𝑥,𝑔,𝑑)

Proof of Theorem bnj1234
StepHypRef Expression
1 fneq1 6470 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑑𝑔 Fn 𝑑))
2 fveq1 6716 . . . . . . 7 (𝑓 = 𝑔 → (𝑓𝑥) = (𝑔𝑥))
3 reseq1 5845 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)))
43opeq2d 4791 . . . . . . . . 9 (𝑓 = 𝑔 → ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5 bnj1234.2 . . . . . . . . 9 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
6 bnj1234.4 . . . . . . . . 9 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
74, 5, 63eqtr4g 2803 . . . . . . . 8 (𝑓 = 𝑔𝑌 = 𝑍)
87fveq2d 6721 . . . . . . 7 (𝑓 = 𝑔 → (𝐺𝑌) = (𝐺𝑍))
92, 8eqeq12d 2753 . . . . . 6 (𝑓 = 𝑔 → ((𝑓𝑥) = (𝐺𝑌) ↔ (𝑔𝑥) = (𝐺𝑍)))
109ralbidv 3118 . . . . 5 (𝑓 = 𝑔 → (∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌) ↔ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
111, 10anbi12d 634 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))))
1211rexbidv 3216 . . 3 (𝑓 = 𝑔 → (∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))))
1312cbvabv 2811 . 2 {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))} = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
14 bnj1234.3 . 2 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
15 bnj1234.5 . 2 𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
1613, 14, 153eqtr4i 2775 1 𝐶 = 𝐷
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  {cab 2714  wral 3061  wrex 3062  cop 4547  cres 5553   Fn wfn 6375  cfv 6380   predc-bnj14 32379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  bnj1245  32707  bnj1256  32708  bnj1259  32709  bnj1296  32714  bnj1311  32717
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