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Theorem fvmpt 5192
 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 = (x 𝐷B)
fvmpt.3 𝐶 V
Assertion
Ref Expression
fvmpt (A 𝐷 → (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 = (x 𝐷B)
42, 3fvmptg 5191 . 2 ((A 𝐷 𝐶 V) → (𝐹A) = 𝐶)
51, 4mpan2 401 1 (A 𝐷 → (𝐹A) = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ↦ cmpt 3809  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853 This theorem is referenced by:  reldm  5754  rdg0  5914  oacl  5979  xpcomco  6236  uzval  8251  sqrtrval  9209
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