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Theorem elrnmpt2 5556
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 5553 . . 3 ran 𝐹 = {zx A y B z = 𝐶}
32eleq2i 2101 . 2 (𝐷 ran 𝐹𝐷 {zx A y B z = 𝐶})
4 elrnmpt2.1 . . . . . 6 𝐶 V
5 eleq1 2097 . . . . . 6 (𝐷 = 𝐶 → (𝐷 V ↔ 𝐶 V))
64, 5mpbiri 157 . . . . 5 (𝐷 = 𝐶𝐷 V)
76rexlimivw 2423 . . . 4 (y B 𝐷 = 𝐶𝐷 V)
87rexlimivw 2423 . . 3 (x A y B 𝐷 = 𝐶𝐷 V)
9 eqeq1 2043 . . . 4 (z = 𝐷 → (z = 𝐶𝐷 = 𝐶))
1092rexbidv 2343 . . 3 (z = 𝐷 → (x A y B z = 𝐶x A y B 𝐷 = 𝐶))
118, 10elab3 2688 . 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 1242   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551  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-ral 2305  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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