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Theorem elrnmpog 5963
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 2177 . . 3 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
212rexbidv 2495 . 2 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
3 rngop.1 . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
43rnmpo 5961 . 2 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
52, 4elab2g 2877 1 (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  wcel 2141  wrex 2449  ran crn 4610  cmpo 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620  df-oprab 5855  df-mpo 5856
This theorem is referenced by:  txopn  13024  xmettxlem  13268  xmettx  13269
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