Theorem ovmpos 5894

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
ovmpos.3	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
ovmpos	⊢ ((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 ovmpos

Step	Hyp	Ref	Expression
1		elex 2697	. . 3 ⊢ (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V)
2		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦𝐵
5		nfcsb1v 3035	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅
6	5	nfel1 2292	. . . . . 6 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅 ∈ V
7		ovmpos.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 5838	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2278	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10		nfcv 2281	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 5801	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2289	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅
13	6, 12	nfim 1551	. . . . 5 ⊢ Ⅎ𝑥(⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)
14		nfcsb1v 3035	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
15	14	nfel1 2292	. . . . . 6 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V
16		nfmpo2 5839	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2278	. . . . . . . 8 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 5801	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2289	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
20	15, 19	nfim 1551	. . . . 5 ⊢ Ⅎ𝑦(⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
21		csbeq1a 3012	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
22	21	eleq1d 2208	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝑅 ∈ V))
23		oveq1 5781	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2154	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅))
25	22, 24	imbi12d 233	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)))
26		csbeq1a 3012	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
27	26	eleq1d 2208	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
28		oveq2 5782	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2154	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
30	27, 29	imbi12d 233	. . . . 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 5893	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1183	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2753	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
34		csbcomg 3025	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
35	34	eleq1d 2208	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
36	34	eqeq2d 2151	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
37	33, 35, 36	3imtr4d 202	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅))
38	1, 37	syl5 32	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅))
39	38	3impia 1178	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⦋csb 3003  (class class class)co 5774   ∈ cmpo 5776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779

This theorem is referenced by: (None)