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Theorem fvmpt2 5334
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 2924 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 2928 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2133 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2225 . . . 4 𝑦𝐵
6 nfcsb1v 2951 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 2929 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 3901 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2105 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 5328 1 ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  csb 2921  cmpt 3868  cfv 4972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-iota 4937  df-fun 4974  df-fv 4980
This theorem is referenced by:  fvmptssdm  5335  fvmpt2d  5337  fvmptdf  5338  mpteqb  5341  fvmptt  5342  fvmptf  5343  ralrnmpt  5389  rexrnmpt  5390  fmptco  5409  f1mpt  5493  offval2  5808  ofrfval2  5809  dom2lem  6422  mapxpen  6497  xpmapenlem  6498  fisumcvg  10588
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