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Theorem ercpbllem 17539
Description: Lemma for ercpbl 17540. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
Hypotheses
Ref Expression
ercpbl.r (𝜑 Er 𝑉)
ercpbl.v (𝜑𝑉𝑊)
ercpbl.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
ercpbllem.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
ercpbllem (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 𝐵))
Distinct variable groups:   𝑥,   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝜑,𝑥
Allowed substitution hints:   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem ercpbllem
StepHypRef Expression
1 ercpbl.r . . . 4 (𝜑 Er 𝑉)
2 ercpbl.v . . . 4 (𝜑𝑉𝑊)
3 ercpbl.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
41, 2, 3divsfval 17538 . . 3 (𝜑 → (𝐹𝐴) = [𝐴] )
51, 2, 3divsfval 17538 . . 3 (𝜑 → (𝐹𝐵) = [𝐵] )
64, 5eqeq12d 2744 . 2 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ [𝐴] = [𝐵] ))
7 ercpbllem.1 . . 3 (𝜑𝐴𝑉)
81, 7erth 8783 . 2 (𝜑 → (𝐴 𝐵 ↔ [𝐴] = [𝐵] ))
96, 8bitr4d 281 1 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098   class class class wbr 5152  cmpt 5235  cfv 6553   Er wer 8730  [cec 8731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-er 8733  df-ec 8735
This theorem is referenced by:  ercpbl  17540  erlecpbl  17541
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