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Theorem fvmpt 5498
Description: Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
Hypotheses
Ref Expression
fvmptg.1 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptg.2 𝐹 = (𝑥𝐷𝐵)
fvmpt.3 𝐶 ∈ V
Assertion
Ref Expression
fvmpt (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt
StepHypRef Expression
1 fvmpt.3 . 2 𝐶 ∈ V
2 fvmptg.1 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
3 fvmptg.2 . . 3 𝐹 = (𝑥𝐷𝐵)
42, 3fvmptg 5497 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
51, 4mpan2 421 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686  cmpt 3989  cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131
This theorem is referenced by:  reldm  6084  rdg0  6284  oacl  6356  fvmptmap  6579  xpcomco  6720  uzval  9340  sqrtrval  10784  fsumcnv  11218  ege2le3  11389  qnumval  11874  qdenval  11875  peano4nninf  13307  peano3nninf  13308  nninfsellemeq  13317
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