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Theorem elrnmpt2g 5532
 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 = (x A, y B𝐶)
Assertion
Ref Expression
elrnmpt2g (𝐷 𝑉 → (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)   𝑉(x,y)

Proof of Theorem elrnmpt2g
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2024 . . 3 (z = 𝐷 → (z = 𝐶𝐷 = 𝐶))
212rexbidv 2323 . 2 (z = 𝐷 → (x A y B z = 𝐶x A y B 𝐷 = 𝐶))
3 rngop.1 . . 3 𝐹 = (x A, y B𝐶)
43rnmpt2 5530 . 2 ran 𝐹 = {zx A y B z = 𝐶}
52, 4elab2g 2662 1 (𝐷 𝑉 → (𝐷 ran 𝐹x A y B 𝐷 = 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370  ∃wrex 2281  ran crn 4269   ↦ cmpt2 5434 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-cnv 4276  df-dm 4278  df-rn 4279  df-oprab 5436  df-mpt2 5437 This theorem is referenced by: (None)
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