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Theorem fvmpt2i 6444
Description: Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypothesis
Ref Expression
mptrcl.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmpt2i (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt2i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3669 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3674 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2802 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 mptrcl.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2894 . . . 4 𝑦𝐵
6 nfcsb1v 3682 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3675 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4893 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2774 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmpti 6435 1 (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  csb 3666  cmpt 4873   I cid 5165  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fv 6049
This theorem is referenced by:  fvmpt2  6445  sumfc  14631  fsumf1o  14645  sumss  14646  isumshft  14762  prodfc  14866  fprodf1o  14867  mbfsup  23622  itg2splitlem  23706  dgrle  24190
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