Theorem ovmpt2s 5547

Description: Value of a function given by the "maps to" notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypothesis

Ref	Expression
ovmpt2s.3	⊢ 𝐹 = (x ∈ 𝐶, y ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
ovmpt2s	⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷 ∧ ⦋A / x⦌⦋B / y⦌𝑅 ∈ 𝑉) → (A𝐹B) = ⦋A / x⦌⦋B / y⦌𝑅)

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

Allowed substitution hints:   𝑅(x,y)   𝐹(x,y)   𝑉(x,y)

Proof of Theorem ovmpt2s

Step	Hyp	Ref	Expression
1		elex 2543	. . 3 ⊢ (⦋A / x⦌⦋B / y⦌𝑅 ∈ 𝑉 → ⦋A / x⦌⦋B / y⦌𝑅 ∈ V)
2		nfcv 2160	. . . . 5 ⊢ ℲxA
3		nfcv 2160	. . . . 5 ⊢ ℲyA
4		nfcv 2160	. . . . 5 ⊢ ℲyB
5		nfcsb1v 2859	. . . . . . 7 ⊢ Ⅎx⦋A / x⦌𝑅
6	5	nfel1 2170	. . . . . 6 ⊢ Ⅎx⦋A / x⦌𝑅 ∈ V
7		ovmpt2s.3	. . . . . . . . 9 ⊢ 𝐹 = (x ∈ 𝐶, y ∈ 𝐷 ↦ 𝑅)
8		nfmpt21 5494	. . . . . . . . 9 ⊢ Ⅎx(x ∈ 𝐶, y ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2157	. . . . . . . 8 ⊢ Ⅎx𝐹
10		nfcv 2160	. . . . . . . 8 ⊢ Ⅎxy
11	2, 9, 10	nfov 5459	. . . . . . 7 ⊢ Ⅎx(A𝐹y)
12	11, 5	nfeq 2167	. . . . . 6 ⊢ Ⅎx(A𝐹y) = ⦋A / x⦌𝑅
13	6, 12	nfim 1446	. . . . 5 ⊢ Ⅎx(⦋A / x⦌𝑅 ∈ V → (A𝐹y) = ⦋A / x⦌𝑅)
14		nfcsb1v 2859	. . . . . . 7 ⊢ Ⅎy⦋B / y⦌⦋A / x⦌𝑅
15	14	nfel1 2170	. . . . . 6 ⊢ Ⅎy⦋B / y⦌⦋A / x⦌𝑅 ∈ V
16		nfmpt22 5495	. . . . . . . . 9 ⊢ Ⅎy(x ∈ 𝐶, y ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2157	. . . . . . . 8 ⊢ Ⅎy𝐹
18	3, 17, 4	nfov 5459	. . . . . . 7 ⊢ Ⅎy(A𝐹B)
19	18, 14	nfeq 2167	. . . . . 6 ⊢ Ⅎy(A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅
20	15, 19	nfim 1446	. . . . 5 ⊢ Ⅎy(⦋B / y⦌⦋A / x⦌𝑅 ∈ V → (A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅)
21		csbeq1a 2837	. . . . . . 7 ⊢ (x = A → 𝑅 = ⦋A / x⦌𝑅)
22	21	eleq1d 2088	. . . . . 6 ⊢ (x = A → (𝑅 ∈ V ↔ ⦋A / x⦌𝑅 ∈ V))
23		oveq1 5443	. . . . . . 7 ⊢ (x = A → (x𝐹y) = (A𝐹y))
24	23, 21	eqeq12d 2036	. . . . . 6 ⊢ (x = A → ((x𝐹y) = 𝑅 ↔ (A𝐹y) = ⦋A / x⦌𝑅))
25	22, 24	imbi12d 223	. . . . 5 ⊢ (x = A → ((𝑅 ∈ V → (x𝐹y) = 𝑅) ↔ (⦋A / x⦌𝑅 ∈ V → (A𝐹y) = ⦋A / x⦌𝑅)))
26		csbeq1a 2837	. . . . . . 7 ⊢ (y = B → ⦋A / x⦌𝑅 = ⦋B / y⦌⦋A / x⦌𝑅)
27	26	eleq1d 2088	. . . . . 6 ⊢ (y = B → (⦋A / x⦌𝑅 ∈ V ↔ ⦋B / y⦌⦋A / x⦌𝑅 ∈ V))
28		oveq2 5444	. . . . . . 7 ⊢ (y = B → (A𝐹y) = (A𝐹B))
29	28, 26	eqeq12d 2036	. . . . . 6 ⊢ (y = B → ((A𝐹y) = ⦋A / x⦌𝑅 ↔ (A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅))
30	27, 29	imbi12d 223	. . . . 5 ⊢ (y = B → ((⦋A / x⦌𝑅 ∈ V → (A𝐹y) = ⦋A / x⦌𝑅) ↔ (⦋B / y⦌⦋A / x⦌𝑅 ∈ V → (A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅)))
31	7	ovmpt4g 5546	. . . . . 6 ⊢ ((x ∈ 𝐶 ∧ y ∈ 𝐷 ∧ 𝑅 ∈ V) → (x𝐹y) = 𝑅)
32	31	3expia 1092	. . . . 5 ⊢ ((x ∈ 𝐶 ∧ y ∈ 𝐷) → (𝑅 ∈ V → (x𝐹y) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2597	. . . 4 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (⦋B / y⦌⦋A / x⦌𝑅 ∈ V → (A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅))
34		csbcomg 2850	. . . . 5 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → ⦋A / x⦌⦋B / y⦌𝑅 = ⦋B / y⦌⦋A / x⦌𝑅)
35	34	eleq1d 2088	. . . 4 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (⦋A / x⦌⦋B / y⦌𝑅 ∈ V ↔ ⦋B / y⦌⦋A / x⦌𝑅 ∈ V))
36	34	eqeq2d 2033	. . . 4 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → ((A𝐹B) = ⦋A / x⦌⦋B / y⦌𝑅 ↔ (A𝐹B) = ⦋B / y⦌⦋A / x⦌𝑅))
37	33, 35, 36	3imtr4d 192	. . 3 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (⦋A / x⦌⦋B / y⦌𝑅 ∈ V → (A𝐹B) = ⦋A / x⦌⦋B / y⦌𝑅))
38	1, 37	syl5 28	. 2 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (⦋A / x⦌⦋B / y⦌𝑅 ∈ 𝑉 → (A𝐹B) = ⦋A / x⦌⦋B / y⦌𝑅))
39	38	3impia 1087	1 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷 ∧ ⦋A / x⦌⦋B / y⦌𝑅 ∈ 𝑉) → (A𝐹B) = ⦋A / x⦌⦋B / y⦌𝑅)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 873   = wceq 1228   ∈ wcel 1374  Vcvv 2535  ⦋csb 2829  (class class class)co 5436   ↦ cmpt2 5438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441

This theorem is referenced by: (None)