Theorem fvmpt2f 6447

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 3678	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3683	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	syl6eq 2811	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		fvmpt2f.0	. . 3 ⊢ Ⅎ𝑥𝐴
5		nfcv 2903	. . 3 ⊢ Ⅎ𝑦𝐴
6		nfcv 2903	. . 3 ⊢ Ⅎ𝑦𝐵
7		nfcsb1v 3691	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		csbeq1a 3684	. . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	4, 5, 6, 7, 8	cbvmptf 4901	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmptg 6444	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2140  Ⅎwnfc 2890  ⦋csb 3675   ↦ cmpt 4882  ‘cfv 6050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058

This theorem is referenced by:  offval2f  7076  fmptcof2  29788  funcnvmptOLD  29798  funcnvmpt  29799  esumc  30444