Theorem fvmpt2 5786

Description: Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)

Hypothesis

Ref	Expression
fvmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fvmpt2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fvmpt2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3150	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3155	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	eqtrdi 2287	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		fvmpt2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		nfcv 2392	. . . 4 ⊢ Ⅎ𝑦𝐵
6		nfcsb1v 3180	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
7		csbeq1a 3156	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
8	5, 6, 7	cbvmpt 4224	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
9	4, 8	eqtri 2259	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmptg 5778	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ⦋csb 3147   ↦ cmpt 4190  ‘cfv 5375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383

This theorem is referenced by:  fvmptssdm  5787  fvmpt2d  5789  fvmptdf  5790  mpteqb  5793  fvmptt  5794  fvmptf  5795  fnmptfvd  5807  ralrnmpt  5844  rexrnmpt  5845  fmptco  5868  f1mpt  5971  offval2  6312  ofrfval2  6313  mptelixpg  7010  dom2lem  7052  mapxpen  7142  xpmapenlem  7143  mkvprop  7492  cc2lem  7626  cc3  7628  fsum3cvg  12128  summodclem2a  12131  fsumf1o  12140  fsum3cvg2  12144  fsumadd  12156  isummulc2  12176  fproddccvg  12322  fprodf1o  12338  prdsbas3  14170  txcnp  15355  cnmpt11  15367  cnmpt1t  15369  elplyd  15825  dvply1  15849  lgseisenlem2  16173