Theorem ovmpos 6155

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 2815	. . 3 ⊢ (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V)
2		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑦𝐵
5		nfcsb1v 3161	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅
6	5	nfel1 2386	. . . . . 6 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅 ∈ V
7		ovmpos.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 6098	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2372	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 6058	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2383	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅
13	6, 12	nfim 1621	. . . . 5 ⊢ Ⅎ𝑥(⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)
14		nfcsb1v 3161	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
15	14	nfel1 2386	. . . . . 6 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V
16		nfmpo2 6099	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2372	. . . . . . . 8 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 6058	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2383	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
20	15, 19	nfim 1621	. . . . 5 ⊢ Ⅎ𝑦(⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
21		csbeq1a 3137	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
22	21	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝑅 ∈ V))
23		oveq1 6035	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2246	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = 𝑅 ↔ (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅))
25	22, 24	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅) ↔ (⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)))
26		csbeq1a 3137	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
27	26	eleq1d 2300	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
28		oveq2 6036	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2246	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
30	27, 29	imbi12d 234	. . . . 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 6154	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1232	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2872	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
34		csbcomg 3151	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
35	34	eleq1d 2300	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
36	34	eqeq2d 2243	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ↔ (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
37	33, 35, 36	3imtr4d 203	. . 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 1227	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ⦋csb 3128  (class class class)co 6028   ∈ cmpo 6030

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033

This theorem is referenced by: (None)