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Theorem ercpbllem 17503
Description: Lemma for ercpbl 17504. (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 17502 . . 3 (𝜑 → (𝐹𝐴) = [𝐴] )
51, 2, 3divsfval 17502 . . 3 (𝜑 → (𝐹𝐵) = [𝐵] )
64, 5eqeq12d 2742 . 2 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ [𝐴] = [𝐵] ))
7 ercpbllem.1 . . 3 (𝜑𝐴𝑉)
81, 7erth 8754 . 2 (𝜑 → (𝐴 𝐵 ↔ [𝐴] = [𝐵] ))
96, 8bitr4d 282 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 5141  cmpt 5224  cfv 6537   Er wer 8702  [cec 8703
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-er 8705  df-ec 8707
This theorem is referenced by:  ercpbl  17504  erlecpbl  17505
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