MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmpt2i Structured version   Visualization version   GIF version

Theorem fvmpt2i 7008
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 3896 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3906 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2eqtrdi 2787 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 mptrcl.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2902 . . . 4 𝑦𝐵
6 nfcsb1v 3918 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3907 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 5259 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2759 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmpti 6997 1 (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  csb 3893  cmpt 5231   I cid 5573  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  fvmpt2  7009  sumfc  15662  fsumf1o  15676  sumss  15677  isumshft  15792  prodfc  15896  fprodf1o  15897  mbfsup  25514  itg2splitlem  25599  dgrle  26096
  Copyright terms: Public domain W3C validator