Theorem fvmpti 7023

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 7022	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶)
4		fvi 6992	. . . 4 ⊢ (𝐶 ∈ V → ( I ‘𝐶) = 𝐶)
5	4	adantl 487	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → ( I ‘𝐶) = 𝐶)
6	3, 5	eqtr4d 2801	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶))
7	1	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
8	2	dmmpt 6260	. . . . . . . 8 ⊢ dom 𝐹 = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ V}
9	7, 8	elrab2 3652	. . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V))
10	9	baib 545	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ dom 𝐹 ↔ 𝐶 ∈ V))
11	10	notbid 321	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V))
12		ndmfv 6948	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
13	11, 12	biimtrrdi 257	. . . 4 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅))
14	13	imp 412	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅)
15		fvprc 6908	. . . 4 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅)
16	15	adantl 487	. . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅)
17	14, 16	eqtr4d 2801	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶))
18	6, 17	pm2.61dan 825	1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281   ↦ cmpt 5192   I cid 5552  dom cdm 5658  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574

This theorem is used by:  fvmpt2i  7035  fvmptex  7039  sumeq2ii  15904  summolem3  15924  fsumf1o  15933  isumshft  16052  prodeq2ii  16124  prodmolem3  16144  fprodf1o  16157