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Theorem ercpbllem 17594
Description: Lemma for ercpbl 17595. (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 17593 . . 3 (𝜑 → (𝐹𝐴) = [𝐴] )
51, 2, 3divsfval 17593 . . 3 (𝜑 → (𝐹𝐵) = [𝐵] )
64, 5eqeq12d 2752 . 2 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ [𝐴] = [𝐵] ))
7 ercpbllem.1 . . 3 (𝜑𝐴𝑉)
81, 7erth 8797 . 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 206   = wceq 1539  wcel 2107   class class class wbr 5142  cmpt 5224  cfv 6560   Er wer 8743  [cec 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-er 8746  df-ec 8748
This theorem is referenced by:  ercpbl  17595  erlecpbl  17596
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