Theorem fvmpts 7018

Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
fvmpts.1	⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵)

Assertion

Ref	Expression
fvmpts	⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpts

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3910	. 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
2		fvmpts.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵)
3		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3932	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3921	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5258	. . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2762	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmptg 7013	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ⦋csb 3907   ↦ cmpt 5230  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570

This theorem is referenced by:  fvmptdf  7021  fvmpocurryd  8294  mptnn0fsupp  14034  mptnn0fsuppr  14036  zsum  15750  prodss  15979  fprodser  15981  fprodn0  16011  fprodefsum  16127  pcmpt  16925  issubc  17885  gsummptnn0fz  20018  mptscmfsupp0  20941  gsummoncoe1  22327  fvmptnn04if  22870  prdsdsf  24392  itgparts  26102  dchrisumlema  27546  abfmpeld  32670  abfmpel  32671  cdlemk40  40899  deg1gprod  42121  aomclem6  43047  ellimcabssub0  45572  constlimc  45579  vonn0ioo2  46645  vonn0icc2  46647