Theorem fvmptt 5653

Description: Closed theorem form of fvmpt 5638. (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 5560	. 2 ⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
3		risset 2525	. . . . 5 ⊢ (𝐴 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐷 𝑥 = 𝐴)
4		elex 2774	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
5		nfa1 1555	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)
6		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐶 ∈ V
7		nffvmpt1 5569	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)
8	7	nfeq1 2349	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶
9	6, 8	nfim 1586	. . . . . . 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 2273	. . . . . . . . . . . . 13 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
14		eqid 2196	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
15	14	fvmpt2 5645	. . . . . . . . . . . . 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 5562	. . . . . . . . . . . 12 ⊢ (((𝑥 = 𝐴 ∧ 𝐵 = 𝐶) ∧ (𝑥 ∈ 𝐷 ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
19	16, 18, 11	3eqtr3d 2237	. . . . . . . . . . 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 1551	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ 𝐷 → (𝑥 = 𝐴 → (𝐶 ∈ V → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))))
24	5, 9, 23	rexlimd 2611	. . . . . 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 2229	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 2167  ∃wrex 2476  Vcvv 2763   ↦ cmpt 4094  ‘cfv 5258

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266

This theorem is referenced by: (None)