Theorem ovmpt2dxf 6786

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

Hypotheses

Ref	Expression
ovmpt2dx.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
ovmpt2dx.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2dx.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
ovmpt2dx.4	⊢ (𝜑 → 𝐴 ∈ 𝐶)
ovmpt2dx.5	⊢ (𝜑 → 𝐵 ∈ 𝐿)
ovmpt2dx.6	⊢ (𝜑 → 𝑆 ∈ 𝑋)
ovmpt2dxf.px	⊢ Ⅎ𝑥𝜑
ovmpt2dxf.py	⊢ Ⅎ𝑦𝜑
ovmpt2dxf.ay	⊢ Ⅎ𝑦𝐴
ovmpt2dxf.bx	⊢ Ⅎ𝑥𝐵
ovmpt2dxf.sx	⊢ Ⅎ𝑥𝑆
ovmpt2dxf.sy	⊢ Ⅎ𝑦𝑆

Assertion

Ref	Expression
ovmpt2dxf	⊢ (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 ovmpt2dxf

Step	Hyp	Ref	Expression
1		ovmpt2dx.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2	1	oveqd 6667	. 2 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
3		ovmpt2dx.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
4		ovmpt2dxf.px	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		ovmpt2dx.5	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐿)
6		ovmpt2dxf.py	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
7		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
8	7	ovmpt4g 6783	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
10	6, 9	alrimi 2082	. . . . . 6 ⊢ (𝜑 → ∀𝑦((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
11	5, 10	spsbcd 3449	. . . . 5 ⊢ (𝜑 → [𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
12	4, 11	alrimi 2082	. . . 4 ⊢ (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
13	3, 12	spsbcd 3449	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥][𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ V) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
14	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐿)
15		simplr 792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
16	3	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝐶)
17	15, 16	eqeltrd 2701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
18	5	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ∈ 𝐿)
19		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20		ovmpt2dx.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
21	20	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐷 = 𝐿)
22	18, 19, 21	3eltr4d 2716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
23		ovmpt2dx.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
24	23	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
25		ovmpt2dx.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
26		elex 3212	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑋 → 𝑆 ∈ V)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ V)
28	27	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ V)
29	24, 28	eqeltrd 2701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ V)
30		biimt 350	. . . . . . 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 1326	. . . . . 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 6668	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
33	32, 24	eqeq12d 2637	. . . . . 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 270	. . . . 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		ovmpt2dxf.ay	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
36	35	nfeq2 2780	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝐴
37	6, 36	nfan 1828	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
38		nfmpt22 6723	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
39		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
40	35, 38, 39	nfov 6676	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
41		ovmpt2dxf.sy	. . . . . . 7 ⊢ Ⅎ𝑦𝑆
42	40, 41	nfeq 2776	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
43	42	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
44	14, 34, 37, 43	sbciedf 3471	. . . 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 2764	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
46		nfmpt21 6722	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
47		ovmpt2dxf.bx	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
48	45, 46, 47	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
49		ovmpt2dxf.sx	. . . . . 6 ⊢ Ⅎ𝑥𝑆
50	48, 49	nfeq 2776	. . . . 5 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
51	50	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
52	3, 44, 4, 51	sbciedf 3471	. . 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 222	. 2 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
54	2, 53	eqtrd 2656	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  Vcvv 3200  [wsbc 3435  (class class class)co 6650   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt2dx  6787  elovmpt2rab  6880  elovmpt2rab1  6881  ovmpt3rab1  6891  mpt2xopoveq  7345  fvmpt2curryd  7397  mdetralt2  20415  mdetunilem2  20419  gsummatr01lem4  20464