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Theorem fvmpt2 5601
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 3062 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3067 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2eqtrdi 2226 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2319 . . . 4 𝑦𝐵
6 nfcsb1v 3092 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3068 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4100 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2198 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 5594 1 ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  csb 3059  cmpt 4066  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226
This theorem is referenced by:  fvmptssdm  5602  fvmpt2d  5604  fvmptdf  5605  mpteqb  5608  fvmptt  5609  fvmptf  5610  fnmptfvd  5622  ralrnmpt  5660  rexrnmpt  5661  fmptco  5684  f1mpt  5774  offval2  6100  ofrfval2  6101  mptelixpg  6736  dom2lem  6774  mapxpen  6850  xpmapenlem  6851  mkvprop  7158  cc2lem  7267  cc3  7269  fsum3cvg  11388  summodclem2a  11391  fsumf1o  11400  fsum3cvg2  11404  fsumadd  11416  isummulc2  11436  fproddccvg  11582  fprodf1o  11598  txcnp  13810  cnmpt11  13822  cnmpt1t  13824  lgseisenlem2  14490
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