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Theorem elrnmpt2g 5555
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 2043 . . 3 (z = 𝐷 → (z = 𝐶𝐷 = 𝐶))
212rexbidv 2343 . 2 (z = 𝐷 → (x A y B z = 𝐶x A y B 𝐷 = 𝐶))
3 rngop.1 . . 3 𝐹 = (x A, y B𝐶)
43rnmpt2 5553 . 2 ran 𝐹 = {zx A y B z = 𝐶}
52, 4elab2g 2683 1 (𝐷 𝑉 → (𝐷 ran 𝐹x A y B 𝐷 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wrex 2301  ran crn 4289  cmpt2 5457
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299  df-oprab 5459  df-mpt2 5460
This theorem is referenced by: (None)
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