Theorem fvmpts 6945

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 3841	. 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
2		fvmpts.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵)
3		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3862	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3852	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5188	. . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2760	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmptg 6939	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⦋csb 3838   ↦ cmpt 5167  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  fvmptdf  6948  fvmpocurryd  8214  mptnn0fsupp  13950  mptnn0fsuppr  13952  zsum  15671  prodss  15903  fprodser  15905  fprodn0  15935  fprodefsum  16051  pcmpt  16854  issubc  17793  gsummptnn0fz  19952  mptscmfsupp0  20913  gsummoncoe1  22283  fvmptnn04if  22824  prdsdsf  24342  itgparts  26024  dchrisumlema  27465  abfmpeld  32742  abfmpel  32743  cdlemk40  41377  deg1gprod  42593  aomclem6  43505  ellimcabssub0  46065  constlimc  46072  vonn0ioo2  47136  vonn0icc2  47138  dftermo4  49989