Theorem fvmptt 5649

Description: Closed theorem form of fvmpt 5634. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Assertion

Ref	Expression
fvmptt	⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶)

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

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

Proof of Theorem fvmptt

Step	Hyp	Ref	Expression
1		simp2 1000	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
2	1	fveq1d 5556	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		risset 2522	. . . . 5 ⊢ (𝐴 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐷 𝑥 = 𝐴)
4		elex 2771	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5		nfa1 1552	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
6		nfv 1539	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐶 ∈ V
7		nffvmpt1 5565	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)
8	7	nfeq1 2346	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶
9	6, 8	nfim 1583	. . . . . . 7 ⊢ Ⅎ𝑥(𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
10		simprl 529	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝑥 ∈ 𝐷)
11		simplr 528	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
12		simprr 531	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
13	11, 12	eqeltrd 2270	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
14		eqid 2193	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
15	14	fvmpt2 5641	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
16	10, 13, 15	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
17		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝑥 = 𝐴)
18	17	fveq2d 5558	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
19	16, 18, 11	3eqtr3d 2234	. . . . . . . . . . 11 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
20	19	exp43 372	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 → (𝑥 ∈ 𝐷 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
21	20	a2i 11	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 = 𝐴 → (𝑥 ∈ 𝐷 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
22	21	com23 78	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
23	22	sps 1548	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
24	5, 9, 23	rexlimd 2608	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (∃𝑥 ∈ 𝐷 𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))
25	4, 24	syl7 69	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (∃𝑥 ∈ 𝐷 𝑥 = 𝐴 → (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))
26	3, 25	biimtrid 152	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝐴 ∈ 𝐷 → (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)))
27	26	imp32 257	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
28	27	3adant2 1018	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
29	2, 28	eqtrd 2226	1 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980  ∀wal 1362   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760   ↦ cmpt 4090  ‘cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262

This theorem is referenced by: (None)