Theorem fvmptt 5628

Description: Closed theorem form of fvmpt 5614. (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 5536	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		risset 2518	. . . . 5 ⊢ (𝐴 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐷 𝑥 = 𝐴)
4		elex 2763	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5		nfa1 1552	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
6		nfv 1539	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐶 ∈ V
7		nffvmpt1 5545	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)
8	7	nfeq1 2342	. . . . . . . 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 2266	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
14		eqid 2189	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
15	14	fvmpt2 5620	. . . . . . . . . . . . 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 5538	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
19	16, 18, 11	3eqtr3d 2230	. . . . . . . . . . 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 2604	. . . . . 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 2222	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 2160  ∃wrex 2469  Vcvv 2752   ↦ cmpt 4079  ‘cfv 5235

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243

This theorem is referenced by: (None)