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Theorem fvmpt 5416
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 5415 . 2 ((𝐴𝐷𝐶 ∈ V) → (𝐹𝐴) = 𝐶)
51, 4mpan2 417 1 (𝐴𝐷 → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1296  wcel 1445  Vcvv 2633  cmpt 3921  cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057
This theorem is referenced by:  reldm  5994  rdg0  6190  oacl  6261  fvmptmap  6482  xpcomco  6622  uzval  9120  sqrtrval  10548  fsumcnv  10980  ege2le3  11110  qnumval  11590  qdenval  11591  peano4nninf  12601  peano3nninf  12602  nninfsellemeq  12611
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