Theorem ovmpt2rdxf 42117

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6786. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
ovmpt2rdx.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
ovmpt2rdx.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpt2rdx.3	⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpt2rdx.4	⊢ (𝜑 → 𝐴 ∈ 𝐿)
ovmpt2rdx.5	⊢ (𝜑 → 𝐵 ∈ 𝐷)
ovmpt2rdx.6	⊢ (𝜑 → 𝑆 ∈ 𝑋)
ovmpt2rdxf.px	⊢ Ⅎ𝑥𝜑
ovmpt2rdxf.py	⊢ Ⅎ𝑦𝜑
ovmpt2rdxf.ay	⊢ Ⅎ𝑦𝐴
ovmpt2rdxf.bx	⊢ Ⅎ𝑥𝐵
ovmpt2rdxf.sx	⊢ Ⅎ𝑥𝑆
ovmpt2rdxf.sy	⊢ Ⅎ𝑦𝑆

Assertion

Ref	Expression
ovmpt2rdxf	⊢ (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 ovmpt2rdxf

Step	Hyp	Ref	Expression
1		ovmpt2rdx.1	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2	1	oveqd 6667	. 2 ⊢ (𝜑 → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
3		ovmpt2rdx.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐿)
4		ovmpt2rdxf.px	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		ovmpt2rdx.5	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
6		ovmpt2rdxf.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 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
10	6, 9	alrimi 2082	. . . . . 6 ⊢ (𝜑 → ∀𝑦((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
11	5, 10	spsbcd 3449	. . . . 5 ⊢ (𝜑 → [𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
12	4, 11	alrimi 2082	. . . 4 ⊢ (𝜑 → ∀𝑥[𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
13	3, 12	spsbcd 3449	. . 3 ⊢ (𝜑 → [𝐴 / 𝑥][𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅))
14	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷)
15	3	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝐿)
16		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
17	16	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
18		ovmpt2rdx.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
19	18	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
20	15, 17, 19	3eltr4d 2716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
21	5	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ∈ 𝐷)
22		eleq1 2689	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
23	22	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑦 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
24	21, 23	mpbird 247	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ∈ 𝐷)
25		ovmpt2rdx.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
26	25	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
27		ovmpt2rdx.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
28	27	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑆 ∈ 𝑋)
29	26, 28	eqeltrd 2701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑅 ∈ 𝑋)
30		biimt 350	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)))
31	20, 24, 29, 30	syl3anc 1326	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅)))
32		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
33	17, 32	oveq12d 6668	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵))
34	33, 26	eqeq12d 2637	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → ((𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
35	31, 34	bitr3d 270	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
36		ovmpt2rdxf.ay	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
37	36	nfeq2 2780	. . . . . 6 ⊢ Ⅎ𝑦 𝑥 = 𝐴
38	6, 37	nfan 1828	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
39		nfmpt22 6723	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
40		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
41	36, 39, 40	nfov 6676	. . . . . . 7 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
42		ovmpt2rdxf.sy	. . . . . . 7 ⊢ Ⅎ𝑦𝑆
43	41, 42	nfeq 2776	. . . . . 6 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
44	43	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
45	14, 35, 38, 44	sbciedf 3471	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
46		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
47		nfmpt21 6722	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
48		ovmpt2rdxf.bx	. . . . . . 7 ⊢ Ⅎ𝑥𝐵
49	46, 47, 48	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)
50		ovmpt2rdxf.sx	. . . . . 6 ⊢ Ⅎ𝑥𝑆
51	49, 50	nfeq 2776	. . . . 5 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆
52	51	a1i 11	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
53	3, 45, 4, 52	sbciedf 3471	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ∧ 𝑅 ∈ 𝑋) → (𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝑦) = 𝑅) ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆))
54	13, 53	mpbid 222	. 2 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)
55	2, 54	eqtrd 2656	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751  [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:  ovmpt2rdx  42118