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Theorem fvmpt2 5511
Description: Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmpt2 ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3009 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3014 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2eqtrdi 2189 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2282 . . . 4 𝑦𝐵
6 nfcsb1v 3039 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3015 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4030 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2161 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 5504 1 ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  csb 3006  cmpt 3996  cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138
This theorem is referenced by:  fvmptssdm  5512  fvmpt2d  5514  fvmptdf  5515  mpteqb  5518  fvmptt  5519  fvmptf  5520  ralrnmpt  5569  rexrnmpt  5570  fmptco  5593  f1mpt  5679  offval2  6004  ofrfval2  6005  mptelixpg  6635  dom2lem  6673  mapxpen  6749  xpmapenlem  6750  mkvprop  7039  cc2lem  7097  cc3  7099  fsum3cvg  11178  summodclem2a  11181  fsumf1o  11190  fsum3cvg2  11194  fsumadd  11206  isummulc2  11226  fproddccvg  11372  txcnp  12477  cnmpt11  12489  cnmpt1t  12491
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