Theorem fvmpt2i 7025

Description: Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypothesis

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

Assertion

Ref	Expression
fvmpt2i	⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem fvmpt2i

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3910	. . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
2		csbid 3920	. . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
3	1, 2	eqtrdi 2790	. 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
4		mptrcl.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦𝐵
6		nfcsb1v 3932	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
7		csbeq1a 3921	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
8	5, 6, 7	cbvmpt 5258	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
9	4, 8	eqtri 2762	. 2 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
10	3, 9	fvmpti 7014	1 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  ⦋csb 3907   ↦ cmpt 5230   I cid 5581  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570

This theorem is referenced by:  fvmpt2  7026  sumfc  15741  fsumf1o  15755  sumss  15756  isumshft  15871  prodfc  15977  fprodf1o  15978  mbfsup  25712  itg2splitlem  25797  dgrle  26296