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Theorem elrnmpog 7287
 Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elrnmpog (𝐷𝑉 → (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)   𝑉(𝑥,𝑦)

Proof of Theorem elrnmpog
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2762 . . 3 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
212rexbidv 3224 . 2 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
3 rngop.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43rnmpo 7285 . 2 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
52, 4elab2g 3591 1 (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  ran crn 5529   ∈ cmpo 7158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-cnv 5536  df-dm 5538  df-rn 5539  df-oprab 7160  df-mpo 7161 This theorem is referenced by:  ordtbas2  21904  txopn  22315  tgisline  26533  elsx  31693  smflimlem6  43810
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