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Theorem fvmpt3 5278
Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
fvmpt3.a (𝑥 = 𝐴𝐵 = 𝐶)
fvmpt3.b 𝐹 = (𝑥𝐷𝐵)
fvmpt3.c (𝑥𝐷𝐵𝑉)
Assertion
Ref Expression
fvmpt3 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt3
StepHypRef Expression
1 fvmpt3.a . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
21eleq1d 2122 . . 3 (𝑥 = 𝐴 → (𝐵𝑉𝐶𝑉))
3 fvmpt3.c . . 3 (𝑥𝐷𝐵𝑉)
42, 3vtoclga 2636 . 2 (𝐴𝐷𝐶𝑉)
5 fvmpt3.b . . 3 𝐹 = (𝑥𝐷𝐵)
61, 5fvmptg 5275 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
74, 6mpdan 406 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  cmpt 3845  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937
This theorem is referenced by:  fvmpt3i  5279  frec2uzzd  9344  frec2uzsucd  9345
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