1stmbfm - Mathbox for Thierry Arnoux

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Theorem 1stmbfm 30322

Description: The first projection map is measurable with regard to the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)

**Hypotheses**
Ref	Expression
1stmbfm.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
1stmbfm.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)

**Assertion**
Ref	Expression
1stmbfm	⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×_s 𝑇)MblFnM𝑆))

Proof of Theorem 1stmbfm Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1stres 7190	. . . 4 ⊢ (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑆
2		1stmbfm.1	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
3		1stmbfm.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
4		sxuni 30256	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×_s 𝑇))
5	2, 3, 4	syl2anc 693	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×_s 𝑇))
6	5	feq2d 6031	. . . 4 ⊢ (𝜑 → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑆 ↔ (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆))
7	1, 6	mpbii 223	. . 3 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆)
8		unielsiga 30191	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
9	2, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
10		sxsiga 30254	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×_s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑆 ×_s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 30191	. . . . 5 ⊢ ((𝑆 ×_s 𝑇) ∈ ∪ ran sigAlgebra → ∪ (𝑆 ×_s 𝑇) ∈ (𝑆 ×_s 𝑇))
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ∪ (𝑆 ×_s 𝑇) ∈ (𝑆 ×_s 𝑇))
14	9, 13	elmapd 7871	. . 3 ⊢ (𝜑 → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑆 ↑_𝑚 ∪ (𝑆 ×_s 𝑇)) ↔ (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆))
15	7, 14	mpbird 247	. 2 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑆 ↑_𝑚 ∪ (𝑆 ×_s 𝑇)))
16		sgon 30187	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
17		sigasspw 30179	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆)
18		pwssb 4612	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝒫 ∪ 𝑆 ↔ ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
19	18	biimpi 206	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝒫 ∪ 𝑆 → ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
20	2, 16, 17, 19	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
21	20	r19.21bi 2932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ⊆ ∪ 𝑆)
22		xpss1 5228	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑆 → (𝑎 × ∪ 𝑇) ⊆ (∪ 𝑆 × ∪ 𝑇))
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 × ∪ 𝑇) ⊆ (∪ 𝑆 × ∪ 𝑇))
24	23	sseld 3602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (𝑎 × ∪ 𝑇) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
25	24	pm4.71rd 667	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇))))
26		ffn 6045	. . . . . . . 8 ⊢ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑆 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
27		elpreima 6337	. . . . . . . 8 ⊢ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎)))
28	1, 26, 27	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎))
29		fvres 6207	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (1^st ‘𝑧))
30	29	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (1^st ‘𝑧) ∈ 𝑎))
31		1st2nd2 7205	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
32		xp2nd 7199	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (2^nd ‘𝑧) ∈ ∪ 𝑇)
33		elxp6 7200	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑎 ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇)))
34		anass 681	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (1^st ‘𝑧) ∈ 𝑎) ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑎 ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇)))
35		an32 839	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (1^st ‘𝑧) ∈ 𝑎) ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) ∧ (1^st ‘𝑧) ∈ 𝑎))
36	33, 34, 35	3bitr2i 288	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) ∧ (1^st ‘𝑧) ∈ 𝑎))
37	36	baib 944	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (1^st ‘𝑧) ∈ 𝑎))
38	31, 32, 37	syl2anc 693	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (1^st ‘𝑧) ∈ 𝑎))
39	30, 38	bitr4d 271	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
40	39	pm5.32i 669	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
41	28, 40	bitri 264	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
42	25, 41	syl6rbbr 279	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
43	42	eqrdv 2620	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (𝑎 × ∪ 𝑇))
44	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
45	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
46		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
47		eqid 2622	. . . . . . . 8 ⊢ ∪ 𝑇 = ∪ 𝑇
48		issgon 30186	. . . . . . . 8 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) ↔ (𝑇 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑇 = ∪ 𝑇))
49	3, 47, 48	sylanblrc 697	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
50		baselsiga 30178	. . . . . . 7 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → ∪ 𝑇 ∈ 𝑇)
51	49, 50	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
52	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∪ 𝑇 ∈ 𝑇)
53		elsx 30257	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) ∧ (𝑎 ∈ 𝑆 ∧ ∪ 𝑇 ∈ 𝑇)) → (𝑎 × ∪ 𝑇) ∈ (𝑆 ×_s 𝑇))
54	44, 45, 46, 52, 53	syl22anc 1327	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 × ∪ 𝑇) ∈ (𝑆 ×_s 𝑇))
55	43, 54	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×_s 𝑇))
56	55	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×_s 𝑇))
57	11, 2	ismbfm 30314	. 2 ⊢ (𝜑 → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×_s 𝑇)MblFnM𝑆) ↔ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑆 ↑_𝑚 ∪ (𝑆 ×_s 𝑇)) ∧ ∀𝑎 ∈ 𝑆 (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×_s 𝑇))))
58	15, 56, 57	mpbir2and 957	1 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×_s 𝑇)MblFnM𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 𝒫 cpw 4158 ⟨cop 4183 ∪ cuni 4436 × cxp 5112 ^◡ccnv 5113 ran crn 5115 ↾ cres 5116 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ↑_𝑚 cmap 7857 sigAlgebracsiga 30170 ×_s csx 30251 MblFnMcmbfm 30312

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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-siga 30171 df-sigagen 30202 df-sx 30252 df-mbfm 30313

This theorem is referenced by: (None)

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