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Theorem elrnmpt2 5614
 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 5611 . . 3 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
32eleq2i 2104 . 2 (𝐷 ∈ ran 𝐹𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶})
4 elrnmpt2.1 . . . . . 6 𝐶 ∈ V
5 eleq1 2100 . . . . . 6 (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
64, 5mpbiri 157 . . . . 5 (𝐷 = 𝐶𝐷 ∈ V)
76rexlimivw 2429 . . . 4 (∃𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
87rexlimivw 2429 . . 3 (∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶𝐷 ∈ V)
9 eqeq1 2046 . . . 4 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
1092rexbidv 2349 . . 3 (𝑧 = 𝐷 → (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶))
118, 10elab3 2694 . 2 (𝐷 ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
123, 11bitri 173 1 (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴𝑦𝐵 𝐷 = 𝐶)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  Vcvv 2557  ran crn 4346   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-oprab 5516  df-mpt2 5517 This theorem is referenced by: (None)
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