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Theorem elrnmpt2 5537
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 = (x A, y B𝐶)
elrnmpt2.1 𝐶 V
Assertion
Ref Expression
elrnmpt2 (𝐷 ran 𝐹x A y B 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 = (x A, y B𝐶)
21rnmpt2 5534 . . 3 ran 𝐹 = {zx A y B z = 𝐶}
32eleq2i 2086 . 2 (𝐷 ran 𝐹𝐷 {zx A y B z = 𝐶})
4 elrnmpt2.1 . . . . . 6 𝐶 V
5 eleq1 2082 . . . . . 6 (𝐷 = 𝐶 → (𝐷 V ↔ 𝐶 V))
64, 5mpbiri 157 . . . . 5 (𝐷 = 𝐶𝐷 V)
76rexlimivw 2407 . . . 4 (y B 𝐷 = 𝐶𝐷 V)
87rexlimivw 2407 . . 3 (x A y B 𝐷 = 𝐶𝐷 V)
9 eqeq1 2028 . . . 4 (z = 𝐷 → (z = 𝐶𝐷 = 𝐶))
1092rexbidv 2327 . . 3 (z = 𝐷 → (x A y B z = 𝐶x A y B 𝐷 = 𝐶))
118, 10elab3 2671 . 2 (𝐷 {zx A y B z = 𝐶} ↔ x A y B 𝐷 = 𝐶)
123, 11bitri 173 1 (𝐷 ran 𝐹x A y B 𝐷 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228   wcel 1374  {cab 2008  wrex 2285  Vcvv 2535  ran crn 4273  cmpt2 5438
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283  df-oprab 5440  df-mpt2 5441
This theorem is referenced by: (None)
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