Theorem ovmpodxf 5967

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovmpodx.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
ovmpodx.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpodx.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpodx.4	⊢ (𝜑 → 𝐴 ∈ 𝐶)
ovmpodx.5	⊢ (𝜑 → 𝐵 ∈ 𝐿)
ovmpodx.6	⊢ (𝜑 → 𝑆 ∈ 𝑋)
ovmpodxf.px	⊢ Ⅎ𝑥𝜑
ovmpodxf.py	⊢ Ⅎ𝑦𝜑
ovmpodxf.ay	⊢ Ⅎ𝑦𝐴
ovmpodxf.bx	⊢ Ⅎ𝑥𝐵
ovmpodxf.sx	⊢ Ⅎ𝑥𝑆
ovmpodxf.sy	⊢ Ⅎ𝑦𝑆

Assertion

Ref	Expression
ovmpodxf	⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpodxf

Step	Hyp	Ref	Expression
1		ovmpodx.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2	1	oveqd 5859	. 2 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
3		ovmpodx.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
4		ovmpodxf.px	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		ovmpodx.5	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐿)
6		ovmpodxf.py	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
7		eqid 2165	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	ovmpt4g 5964	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
9	8	a1i 9	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
10	6, 9	alrimi 1510	. . . . . 6 ⊢ (𝜑 → ∀𝑦((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
11	5, 10	spsbcd 2963	. . . . 5 ⊢ (𝜑 → [𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
12	4, 11	alrimi 1510	. . . 4 ⊢ (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
13	3, 12	spsbcd 2963	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥][𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
14	5	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐿)
15		simplr 520	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
16	3	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝐶)
17	15, 16	eqeltrd 2243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
18	5	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ∈ 𝐿)
19		simpr 109	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20		ovmpodx.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
21	20	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
22	18, 19, 21	3eltr4d 2250	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
23		ovmpodx.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
24	23	anassrs 398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25		ovmpodx.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
26		elex 2737	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑋 → 𝑆 ∈ V)
27	25, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ V)
28	27	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
29	24, 28	eqeltrd 2243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30		biimt 240	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)))
31	17, 22, 29, 30	syl3anc 1228	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)))
32	15, 19	oveq12d 5860	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
33	32, 24	eqeq12d 2180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
34	31, 33	bitr3d 189	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
35		ovmpodxf.ay	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
36	35	nfeq2 2320	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝐴
37	6, 36	nfan 1553	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
38		nfmpo2 5910	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
39		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
40	35, 38, 39	nfov 5872	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
41		ovmpodxf.sy	. . . . . . 7 ⊢ Ⅎ𝑦𝑆
42	40, 41	nfeq 2316	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
43	42	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
44	14, 34, 37, 43	sbciedf 2986	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
45		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
46		nfmpo1 5909	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
47		ovmpodxf.bx	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
48	45, 46, 47	nfov 5872	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
49		ovmpodxf.sx	. . . . . 6 ⊢ Ⅎ𝑥𝑆
50	48, 49	nfeq 2316	. . . . 5 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
51	50	a1i 9	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
52	3, 44, 4, 51	sbciedf 2986	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
53	13, 52	mpbid 146	. 2 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
54	2, 53	eqtrd 2198	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295  Vcvv 2726  [wsbc 2951  (class class class)co 5842   ∈ cmpo 5844

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ovmpodx  5968  mpoxopoveq  6208