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Theorem ercpbllem 17431
Description: Lemma for ercpbl 17432. (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 17430 . . 3 (𝜑 → (𝐹𝐴) = [𝐴] )
51, 2, 3divsfval 17430 . . 3 (𝜑 → (𝐹𝐵) = [𝐵] )
64, 5eqeq12d 2753 . 2 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ [𝐴] = [𝐵] ))
7 ercpbllem.1 . . 3 (𝜑𝐴𝑉)
81, 7erth 8698 . 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 1542  wcel 2107   class class class wbr 5106  cmpt 5189  cfv 6497   Er wer 8646  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-er 8649  df-ec 8651
This theorem is referenced by:  ercpbl  17432  erlecpbl  17433
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