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Theorem fvmpt2 5197
Description: Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (x AB)
Assertion
Ref Expression
fvmpt2 ((x A B 𝐶) → (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 2849 . . 3 (y = xy / xB = x / xB)
2 csbid 2853 . . 3 x / xB = B
31, 2syl6eq 2085 . 2 (y = xy / xB = B)
4 fvmpt2.1 . . 3 𝐹 = (x AB)
5 nfcv 2175 . . . 4 yB
6 nfcsb1v 2876 . . . 4 xy / xB
7 csbeq1a 2854 . . . 4 (x = yB = y / xB)
85, 6, 7cbvmpt 3842 . . 3 (x AB) = (y Ay / xB)
94, 8eqtri 2057 . 2 𝐹 = (y Ay / xB)
103, 9fvmptg 5191 1 ((x A B 𝐶) → (𝐹x) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  csb 2846  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-bnd 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-csb 2847  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:  fvmptssdm  5198  fvmpt2d  5200  fvmptdf  5201  mpteqb  5204  fvmptt  5205  fvmptf  5206  ralrnmpt  5252  rexrnmpt  5253  fmptco  5273  f1mpt  5353  offval2  5668  ofrfval2  5669  dom2lem  6188
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