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Theorem fvmpt2 5254
 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 2855 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 2859 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2088 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 fvmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2178 . . . 4 𝑦𝐵
6 nfcsb1v 2882 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 2860 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 3851 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2060 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmptg 5248 1 ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ⦋csb 2852   ↦ cmpt 3818  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910 This theorem is referenced by:  fvmptssdm  5255  fvmpt2d  5257  fvmptdf  5258  mpteqb  5261  fvmptt  5262  fvmptf  5263  ralrnmpt  5309  rexrnmpt  5310  fmptco  5330  f1mpt  5410  offval2  5726  ofrfval2  5727  dom2lem  6252
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