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Theorem elrnmpt2 6758
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 6755 . . 3 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32eleq2i 2691 . 2 (𝐷 ∈ ran 𝐹𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶})
4 elrnmpt2.1 . . . . . 6 𝐶 ∈ V
5 eleq1 2687 . . . . . 6 (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
64, 5mpbiri 248 . . . . 5 (𝐷 = 𝐶𝐷 ∈ V)
76rexlimivw 3025 . . . 4 (∃𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
87rexlimivw 3025 . . 3 (∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
9 eqeq1 2624 . . . 4 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
1092rexbidv 3053 . . 3 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
118, 10elab3 3352 . 2 (𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
123, 11bitri 264 1 (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1481  wcel 1988  {cab 2606  wrex 2910  Vcvv 3195  ran crn 5105  cmpt2 6637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-cnv 5112  df-dm 5114  df-rn 5115  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  qexALT  11788  lsmelvalx  18036  efgtlen  18120  frgpnabllem1  18257  fmucndlem  22076  mbfimaopnlem  23403  tglnunirn  25424  tpr2rico  29932  mbfmco2  30301  br2base  30305  dya2icobrsiga  30312  dya2iocnrect  30317  dya2iocucvr  30320  sxbrsigalem2  30322  cntotbnd  33566  eldiophb  37139  elicores  39563  volicorescl  40530
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