Theorem fvmptt 5726

Description: Closed theorem form of fvmpt 5711. (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 1022	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))
2	1	fveq1d 5629	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		risset 2558	. . . . 5 ⊢ (𝐴 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐷 𝑥 = 𝐴)
4		elex 2811	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5		nfa1 1587	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
6		nfv 1574	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐶 ∈ V
7		nffvmpt1 5638	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)
8	7	nfeq1 2382	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶
9	6, 8	nfim 1618	. . . . . . 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 2306	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
14		eqid 2229	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
15	14	fvmpt2 5718	. . . . . . . . . . . . 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 5631	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
19	16, 18, 11	3eqtr3d 2270	. . . . . . . . . . 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 1583	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
24	5, 9, 23	rexlimd 2645	. . . . . 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 1040	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
29	2, 28	eqtrd 2262	1 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002  ∀wal 1393   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799   ↦ cmpt 4145  ‘cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326

This theorem is referenced by: (None)