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

Theorem fvmpt2i 6257
 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 3522 . . 3 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
2 csbid 3527 . . 3 𝑥 / 𝑥𝐵 = 𝐵
31, 2syl6eq 2671 . 2 (𝑦 = 𝑥𝑦 / 𝑥𝐵 = 𝐵)
4 mptrcl.1 . . 3 𝐹 = (𝑥𝐴𝐵)
5 nfcv 2761 . . . 4 𝑦𝐵
6 nfcsb1v 3535 . . . 4 𝑥𝑦 / 𝑥𝐵
7 csbeq1a 3528 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
85, 6, 7cbvmpt 4719 . . 3 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
94, 8eqtri 2643 . 2 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
103, 9fvmpti 6248 1 (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ⦋csb 3519   ↦ cmpt 4683   I cid 4994  ‘cfv 5857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865 This theorem is referenced by:  fvmpt2  6258  sumfc  14389  fsumf1o  14403  sumss  14404  isumshft  14515  prodfc  14619  fprodf1o  14620  mbfsup  23371  itg2splitlem  23455  dgrle  23937
 Copyright terms: Public domain W3C validator