Theorem fvmpt2f 6942

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 3852	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3862	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	eqtrdi 2787	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		fvmpt2f.0	. . 3 ⊢ Ⅎ𝑥𝐴
5		nfcv 2898	. . 3 ⊢ Ⅎ𝑦𝐴
6		nfcv 2898	. . 3 ⊢ Ⅎ𝑦𝐵
7		nfcsb1v 3873	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		csbeq1a 3863	. . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	4, 5, 6, 7, 8	cbvmptf 5198	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmptg 6939	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883  ⦋csb 3849   ↦ cmpt 5179  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  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:  offval2f  7637  fmptcof2  32735  funcnvmpt  32745  esumc  34208  fvmpt2df  45512  fvmpt4d  45516  smfpimltxrmptf  46998  smfpimgtxrmptf  47024