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Theorem elrnmpo 7505
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)
elrnmpo.1 𝐶 ∈ V
Assertion
Ref Expression
elrnmpo (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 elrnmpo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21rnmpo 7502 . . 3 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32eleq2i 2820 . 2 (𝐷 ∈ ran 𝐹𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶})
4 elrnmpo.1 . . . . . 6 𝐶 ∈ V
5 eleq1 2816 . . . . . 6 (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
64, 5mpbiri 258 . . . . 5 (𝐷 = 𝐶𝐷 ∈ V)
76rexlimivw 3130 . . . 4 (∃𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
87rexlimivw 3130 . . 3 (∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
9 eqeq1 2733 . . . 4 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
1092rexbidv 3200 . . 3 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
118, 10elab3 3650 . 2 (𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
123, 11bitri 275 1 (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2109  {cab 2707  wrex 3053  Vcvv 3444  ran crn 5632  cmpo 7371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-cnv 5639  df-dm 5641  df-rn 5642  df-oprab 7373  df-mpo 7374
This theorem is referenced by:  qexALT  12899  lsmelvalx  19546  efgtlen  19632  frgpnabllem1  19779  fmucndlem  24154  mbfimaopnlem  25532  tglnunirn  28451  tpr2rico  33875  mbfmco2  34229  br2base  34233  dya2icobrsiga  34240  dya2iocnrect  34245  dya2iocucvr  34248  sxbrsigalem2  34250  cntotbnd  37763  eldiophb  42718  elicores  45504  volicorescl  46524
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