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Theorem fvmpt2i 6199
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 3502 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3507 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2660 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 mptrcl.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2751 . . . 4 𝑦𝐵
6 nfcsb1v 3515 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3508 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4677 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2632 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmpti 6190 1 (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  csb 3499  cmpt 4643   I cid 4948  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812
This theorem is referenced by:  fvmpt2  6200  sumfc  14287  fsumf1o  14301  sumss  14302  isumshft  14410  prodfc  14514  fprodf1o  14515  mbfsup  23237  itg2splitlem  23321  dgrle  23803
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