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Theorem elrnmpt2 6671
 Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
elrnmpt2.1 𝐶 ∈ V
Assertion
Ref Expression
elrnmpt2 (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem elrnmpt2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpt2 6668 . . 3 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32eleq2i 2680 . 2 (𝐷 ∈ ran 𝐹𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶})
4 elrnmpt2.1 . . . . . 6 𝐶 ∈ V
5 eleq1 2676 . . . . . 6 (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
64, 5mpbiri 247 . . . . 5 (𝐷 = 𝐶𝐷 ∈ V)
76rexlimivw 3011 . . . 4 (∃𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
87rexlimivw 3011 . . 3 (∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
9 eqeq1 2614 . . . 4 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
1092rexbidv 3039 . . 3 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
118, 10elab3 3327 . 2 (𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
123, 11bitri 263 1 (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173  ran crn 5039   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  qexALT  11679  lsmelvalx  17878  efgtlen  17962  frgpnabllem1  18099  fmucndlem  21905  mbfimaopnlem  23228  tglnunirn  25243  tpr2rico  29286  mbfmco2  29654  br2base  29658  dya2icobrsiga  29665  dya2iocnrect  29670  dya2iocucvr  29673  sxbrsigalem2  29675  cntotbnd  32765  eldiophb  36338  elicores  38607  volicorescl  39443
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