Theorem fvmpt2f 6241

Description: Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)

Hypothesis

Ref	Expression
fvmpt2f.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
fvmpt2f	⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

Proof of Theorem fvmpt2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3522	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3527	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	syl6eq 2676	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		fvmpt2f.0	. . 3 ⊢ Ⅎ𝑥𝐴
5		nfcv 2767	. . 3 ⊢ Ⅎ𝑦𝐴
6		nfcv 2767	. . 3 ⊢ Ⅎ𝑦𝐵
7		nfcsb1v 3535	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		csbeq1a 3528	. . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	4, 5, 6, 7, 8	cbvmptf 4713	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmptg 6238	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  Ⅎwnfc 2754  ⦋csb 3519   ↦ cmpt 4678  ‘cfv 5850

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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872

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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858

This theorem is referenced by:  offval2f  6863  fmptcof2  29290  funcnvmptOLD  29301  funcnvmpt  29302  esumc  29886