Theorem fvmpts 6999

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 3896	. 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
2		fvmpts.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵)
3		nfcv 2904	. . . 4 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3918	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3907	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5259	. . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2761	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmptg 6994	1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ⦋csb 3893   ↦ cmpt 5231  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-iota 6493  df-fun 6543  df-fv 6549

This theorem is referenced by:  fvmptdf  7002  fvmpocurryd  8253  mptnn0fsupp  13959  mptnn0fsuppr  13961  zsum  15661  prodss  15888  fprodser  15890  fprodn0  15920  fprodefsum  16035  pcmpt  16822  issubc  17782  gsummptnn0fz  19849  mptscmfsupp0  20530  gsummoncoe1  21820  fvmptnn04if  22343  prdsdsf  23865  itgparts  25556  dchrisumlema  26981  abfmpeld  31867  abfmpel  31868  cdlemk40  39777  aomclem6  41787  ellimcabssub0  44320  constlimc  44327  vonn0ioo2  45393  vonn0icc2  45395