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Theorem ercpbllem 16331
 Description: Lemma for ercpbl 16332. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
ercpbl.r (𝜑 Er 𝑉)
ercpbl.v (𝜑𝑉 ∈ V)
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 hint:   𝐹(𝑥)

Proof of Theorem ercpbllem
StepHypRef Expression
1 ercpbl.r . . . 4 (𝜑 Er 𝑉)
2 ercpbl.v . . . 4 (𝜑𝑉 ∈ V)
3 ercpbl.f . . . 4 𝐹 = (𝑥𝑉 ↦ [𝑥] )
41, 2, 3divsfval 16330 . . 3 (𝜑 → (𝐹𝐴) = [𝐴] )
51, 2, 3divsfval 16330 . . 3 (𝜑 → (𝐹𝐵) = [𝐵] )
64, 5eqeq12d 2739 . 2 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ [𝐴] = [𝐵] ))
7 ercpbllem.1 . . 3 (𝜑𝐴𝑉)
81, 7erth 7909 . 2 (𝜑 → (𝐴 𝐵 ↔ [𝐴] = [𝐵] ))
96, 8bitr4d 271 1 (𝜑 → ((𝐹𝐴) = (𝐹𝐵) ↔ 𝐴 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1596   ∈ wcel 2103  Vcvv 3304   class class class wbr 4760   ↦ cmpt 4837  ‘cfv 6001   Er wer 7859  [cec 7860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fv 6009  df-er 7862  df-ec 7864 This theorem is referenced by:  ercpbl  16332  erlecpbl  16333
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