Theorem ovmpos 7489

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 3457	. . 3 ⊢ (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V)
2		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝐵
5		nfcsb1v 3869	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅
6	5	nfel1 2911	. . . . . 6 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅 ∈ V
7		ovmpos.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 7421	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 7371	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2908	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅
13	6, 12	nfim 1897	. . . . 5 ⊢ Ⅎ𝑥(⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)
14		nfcsb1v 3869	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
15	14	nfel1 2911	. . . . . 6 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V
16		nfmpo2 7422	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2892	. . . . . . . 8 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 7371	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2908	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
20	15, 19	nfim 1897	. . . . 5 ⊢ Ⅎ𝑦(⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
21		csbeq1a 3859	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
22	21	eleq1d 2816	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝑅 ∈ V))
23		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅))
25	22, 24	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)))
26		csbeq1a 3859	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
27	26	eleq1d 2816	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
28		oveq2 7349	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2747	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
30	27, 29	imbi12d 344	. . . . 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 7488	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1121	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 3530	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
34		csbcom 4365	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
35	34	eleq1i 2822	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V)
36	34	eqeq2i 2744	. . . 4 ⊢ ((𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
37	33, 35, 36	3imtr4g 296	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅))
38	1, 37	syl5 34	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅))
39	38	3impia 1117	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3845  (class class class)co 7341   ∈ cmpo 7343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346

This theorem is referenced by:  finxpreclem2  37424