Theorem ovmptss 8030

Description: If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
ovmptss.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
ovmptss	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

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

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem ovmptss

Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovmptss.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		mpomptsx 8001	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
3	1, 2	eqtri 2759	. . 3 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
4	3	fvmptss 6965	. 2 ⊢ (∀𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5		vex 3450	. . . . . . . 8 ⊢ 𝑢 ∈ V
6		vex 3450	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	5, 6	op1std 7936	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
8	7	csbeq1d 3862	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
9	5, 6	op2ndd 7937	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
10	9	csbeq1d 3862	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11	10	csbeq2dv 3865	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	8, 11	eqtrd 2771	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	12	sseq1d 3978	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
14	13	raliunxp 5800	. . 3 ⊢ (∀𝑧 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋)
15		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
16		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥{𝑢}
17		nfcsb1v 3883	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
18	16, 17	nfxp 5671	. . . . 5 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
19		sneq 4601	. . . . . 6 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
20		csbeq1a 3872	. . . . . 6 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
21	19, 20	xpeq12d 5669	. . . . 5 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
22	15, 18, 21	cbviun 5001	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
23	22	raleqi 3309	. . 3 ⊢ (∀𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ∀𝑧 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋)
24		nfv 1917	. . . 4 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋
25		nfcsb1v 3883	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
26		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥𝑋
27	25, 26	nfss 3939	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
28	17, 27	nfralw 3292	. . . 4 ⊢ Ⅎ𝑥∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
29		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑣 𝐶 ⊆ 𝑋
30		nfcsb1v 3883	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
31		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
32	30, 31	nfss 3939	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
33		csbeq1a 3872	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
34	33	sseq1d 3978	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐶 ⊆ 𝑋 ↔ ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
35	29, 32, 34	cbvralw 3287	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ 𝐵 ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋)
36		csbeq1a 3872	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
37	36	sseq1d 3978	. . . . . 6 ⊢ (𝑥 = 𝑢 → (⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
38	20, 37	raleqbidv 3317	. . . . 5 ⊢ (𝑥 = 𝑢 → (∀𝑣 ∈ 𝐵 ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
39	35, 38	bitrid 282	. . . 4 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
40	24, 28, 39	cbvralw 3287	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋)
41	14, 23, 40	3bitr4ri 303	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋)
42		df-ov 7365	. . 3 ⊢ (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
43	42	sseq1i 3975	. 2 ⊢ ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
44	4, 41, 43	3imtr4i 291	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1541  ∀wral 3060  ⦋csb 3858   ⊆ wss 3913  {csn 4591  ⟨cop 4597  ∪ ciun 4959   ↦ cmpt 5193   × cxp 5636  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  1^st c1st 7924  2^nd c2nd 7925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927

This theorem is referenced by:  relmpoopab  8031  relxpchom  18083  reldv  25271