Theorem ovmptss 8090

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 8061	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
3	1, 2	eqtri 2783	. . 3 ⊢ 𝐹 = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
4	3	fvmptss 6999	. 2 ⊢ (∀𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5		vex 3454	. . . . . . . 8 ⊢ 𝑢 ∈ V
6		vex 3454	. . . . . . . 8 ⊢ 𝑣 ∈ V
7	5, 6	op1std 7996	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
8	7	csbeq1d 3851	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
9	5, 6	op2ndd 7997	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
10	9	csbeq1d 3851	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11	10	csbeq2dv 3854	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	8, 11	eqtrd 2795	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	12	sseq1d 3962	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
14	13	raliunxp 5819	. . 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 2922	. . . . 5 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
16		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥{𝑢}
17		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
18	16, 17	nfxp 5688	. . . . 5 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
19		sneq 4594	. . . . . 6 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
20		csbeq1a 3861	. . . . . 6 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
21	19, 20	xpeq12d 5686	. . . . 5 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
22	15, 18, 21	cbviun 4993	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
23	22	raleqi 3317	. . 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 1947	. . . 4 ⊢ Ⅎ𝑢∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋
25		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
26		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝑋
27	25, 26	nfss 3924	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
28	17, 27	nfralw 3309	. . . 4 ⊢ Ⅎ𝑥∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
29		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑣 𝐶 ⊆ 𝑋
30		nfcsb1v 3871	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
31		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
32	30, 31	nfss 3924	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋
33		csbeq1a 3861	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
34	33	sseq1d 3962	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝐶 ⊆ 𝑋 ↔ ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
35	29, 32, 34	cbvralw 3304	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ 𝐵 ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋)
36		csbeq1a 3861	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
37	36	sseq1d 3962	. . . . . 6 ⊢ (𝑥 = 𝑢 → (⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
38	20, 37	raleqbidv 3334	. . . . 5 ⊢ (𝑥 = 𝑢 → (∀𝑣 ∈ 𝐵 ⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
39	35, 38	bitrid 286	. . . 4 ⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋))
40	24, 28, 39	cbvralw 3304	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ ⦋ 𝑢 / 𝑥⦌𝐵⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶 ⊆ 𝑋)
41	14, 23, 40	3bitr4ri 307	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 ↔ ∀𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 ⊆ 𝑋)
42		df-ov 7416	. . 3 ⊢ (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
43	42	sseq1i 3959	. 2 ⊢ ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
44	4, 41, 43	3imtr4i 295	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∀wral 3076  ⦋csb 3847   ⊆ wss 3899  {csn 4584  ⟨cop 4590  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5653  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987

This theorem is used by:  relmpoopab  8091  relxpchom  18269  reldv  26097