Theorem fvmpt2f 6943

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 3841	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3851	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	eqtrdi 2791	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		fvmpt2f.0	. . 3 ⊢ Ⅎ𝑥𝐴
5		nfcv 2902	. . 3 ⊢ Ⅎ𝑦𝐴
6		nfcv 2902	. . 3 ⊢ Ⅎ𝑦𝐵
7		nfcsb1v 3862	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		csbeq1a 3852	. . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
9	4, 5, 6, 7, 8	cbvmptf 5179	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmptg 6940	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2887  ⦋csb 3838   ↦ cmpt 5160  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  funcnvmpt  6944  offval2f  7642  fmptcof2  32756  esumc  34242  fvmpt2df  45723  fvmpt4d  45727  smfpimltxrmptf  47208  smfpimgtxrmptf  47234