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Theorem fvmpt2i 6023
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 3388 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3393 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2555 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 mptrcl.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2646 . . . 4 𝑦𝐵
6 nfcsb1v 3401 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3394 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4527 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2527 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmpti 6014 1 (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1468  wcel 1937  csb 3385  cmpt 4493   I cid 4790  cfv 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-mpt 4495  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5597  df-fun 5635  df-fv 5641
This theorem is referenced by:  fvmpt2  6024  sumfc  13935  fsumf1o  13949  sumss  13950  isumshft  14057  prodfc  14159  fprodf1o  14160  mbfsup  22781  itg2splitlem  22867  dgrle  23358
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