Theorem ovmpos 6082

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 2785	. . 3 ⊢ (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V)
2		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑦𝐵
5		nfcsb1v 3130	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅
6	5	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝑅 ∈ V
7		ovmpos.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8		nfmpo1 6025	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
9	7, 8	nfcxfr 2346	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
10		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
11	2, 9, 10	nfov 5987	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴𝐹𝑦)
12	11, 5	nfeq 2357	. . . . . 6 ⊢ Ⅎ𝑥(𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅
13	6, 12	nfim 1596	. . . . 5 ⊢ Ⅎ𝑥(⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝑦) = ⦋𝐴 / 𝑥⦌𝑅)
14		nfcsb1v 3130	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
15	14	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V
16		nfmpo2 6026	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
17	7, 16	nfcxfr 2346	. . . . . . . 8 ⊢ Ⅎ𝑦𝐹
18	3, 17, 4	nfov 5987	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴𝐹𝐵)
19	18, 14	nfeq 2357	. . . . . 6 ⊢ Ⅎ𝑦(𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅
20	15, 19	nfim 1596	. . . . 5 ⊢ Ⅎ𝑦(⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
21		csbeq1a 3106	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
22	21	eleq1d 2275	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑅 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝑅 ∈ V))
23		oveq1 5964	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
24	23, 21	eqeq12d 2221	. . . . . 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 3106	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
27	26	eleq1d 2275	. . . . . 6 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
28		oveq2 5965	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
29	28, 26	eqeq12d 2221	. . . . . 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 6081	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥𝐹𝑦) = 𝑅)
32	31	3expia 1208	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑅 ∈ V → (𝑥𝐹𝑦) = 𝑅))
33	2, 3, 4, 13, 20, 25, 30, 32	vtocl2gaf 2842	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅))
34		csbcomg 3120	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅)
35	34	eleq1d 2275	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ V ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝑅 ∈ V))
36	34	eqeq2d 2218	. . . 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 1203	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  Vcvv 2773  ⦋csb 3097  (class class class)co 5957   ∈ cmpo 5959

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962

This theorem is referenced by: (None)