Theorem fvmpti 7015

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

Hypotheses

Ref	Expression
fvmptg.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvmptg.2	⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)

Assertion

Ref	Expression
fvmpti	⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

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

Proof of Theorem fvmpti

Step	Hyp	Ref	Expression
1		fvmptg.1	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
2		fvmptg.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
3	1, 2	fvmptg 7014	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶)
4		fvi 6985	. . . 4 ⊢ (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
5	4	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
6	3, 5	eqtr4d 2778	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶))
7	1	eleq1d 2824	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
8	2	dmmpt 6262	. . . . . . . 8 ⊢ dom 𝐹 = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ V}
9	7, 8	elrab2 3698	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V))
10	9	baib 535	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ dom 𝐹 ↔ 𝐶 ∈ V))
11	10	notbid 318	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12		ndmfv 6942	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
13	11, 12	biimtrrdi 254	. . . 4 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅))
14	13	imp 406	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅)
15		fvprc 6899	. . . 4 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
16	15	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
17	14, 16	eqtr4d 2778	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶))
18	6, 17	pm2.61dan 813	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339   ↦ cmpt 5231   I cid 5582  dom cdm 5689  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571

This theorem is referenced by:  fvmpt2i  7026  fvmptex  7030  sumeq2ii  15726  summolem3  15747  fsumf1o  15756  isumshft  15872  prodeq2ii  15944  prodmolem3  15966  fprodf1o  15979