1stmbfm - Mathbox for Thierry Arnoux

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Theorem 1stmbfm 34559

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 7996	. . . 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 34492	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×_s 𝑇))
5	2, 3, 4	syl2anc 593	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×_s 𝑇))
6	5	feq2d 6677	. . . 4 ⊢ (𝜑 → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑆 ↔ (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆))
7	1, 6	mpbii 235	. . 3 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆)
8		unielsiga 34427	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
9	2, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
10		sxsiga 34490	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×_s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑆 ×_s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 34427	. . . . 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 8823	. . 3 ⊢ (𝜑 → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑆 ↑_m ∪ (𝑆 ×_s 𝑇)) ↔ (1^st ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×_s 𝑇)⟶∪ 𝑆))
15	7, 14	mpbird 259	. 2 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑆 ↑_m ∪ (𝑆 ×_s 𝑇)))
16		ffn 6693	. . . . . . . 8 ⊢ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑆 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
17		elpreima 7041	. . . . . . . 8 ⊢ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎)))
18	1, 16, 17	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎))
19		fvres 6888	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (1^st ‘𝑧))
20	19	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (1^st ‘𝑧) ∈ 𝑎))
21		1st2nd2 8011	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
22		xp2nd 8005	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (2^nd ‘𝑧) ∈ ∪ 𝑇)
23		elxp6 8006	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑎 ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇)))
24		anass 472	. . . . . . . . . . . 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)))
25		an32 656	. . . . . . . . . . . 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))
26	23, 24, 25	3bitr2i 301	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) ∧ (1^st ‘𝑧) ∈ 𝑎))
27	26	baib 543	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ ∪ 𝑇) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (1^st ‘𝑧) ∈ 𝑎))
28	21, 22, 27	syl2anc 593	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (1^st ‘𝑧) ∈ 𝑎))
29	20, 28	bitr4d 284	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
30	29	pm5.32i 582	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((1^st ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
31	18, 30	bitri 277	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
32		sgon 34423	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
33		sigasspw 34415	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆)
34		pwssb 5060	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝒫 ∪ 𝑆 ↔ ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
35	34	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝒫 ∪ 𝑆 → ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
36	2, 32, 33, 35	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 𝑎 ⊆ ∪ 𝑆)
37	36	r19.21bi 3256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ⊆ ∪ 𝑆)
38		xpss1 5668	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑆 → (𝑎 × ∪ 𝑇) ⊆ (∪ 𝑆 × ∪ 𝑇))
39	37, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 × ∪ 𝑇) ⊆ (∪ 𝑆 × ∪ 𝑇))
40	39	sseld 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (𝑎 × ∪ 𝑇) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
41	40	pm4.71rd 570	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (𝑎 × ∪ 𝑇) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (𝑎 × ∪ 𝑇))))
42	31, 41	bitr4id 292	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑧 ∈ (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (𝑎 × ∪ 𝑇)))
43	42	eqrdv 2762	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (𝑎 × ∪ 𝑇))
44	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
45	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
46		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
47		eqid 2764	. . . . . . . 8 ⊢ ∪ 𝑇 = ∪ 𝑇
48		issgon 34422	. . . . . . . 8 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) ↔ (𝑇 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑇 = ∪ 𝑇))
49	3, 47, 48	sylanblrc 599	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
50		baselsiga 34414	. . . . . . 7 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → ∪ 𝑇 ∈ 𝑇)
51	49, 50	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
52	51	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → ∪ 𝑇 ∈ 𝑇)
53		elsx 34493	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) ∧ (𝑎 ∈ 𝑆 ∧ ∪ 𝑇 ∈ 𝑇)) → (𝑎 × ∪ 𝑇) ∈ (𝑆 ×_s 𝑇))
54	44, 45, 46, 52, 53	syl22anc 849	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (𝑎 × ∪ 𝑇) ∈ (𝑆 ×_s 𝑇))
55	43, 54	eqeltrd 2864	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆) → (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×_s 𝑇))
56	55	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (^◡(1^st ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×_s 𝑇))
57	11, 2	ismbfm 34550	. 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 723	1 ⊢ (𝜑 → (1^st ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×_s 𝑇)MblFnM𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ∀wral 3078 ⊆ wss 3906 𝒫 cpw 4557 ⟨cop 4590 ∪ cuni 4867 × cxp 5647 ^◡ccnv 5648 ran crn 5650 ↾ cres 5651 “ cima 5652 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 1^st c1st 7970 2^nd c2nd 7971 ↑_m cmap 8810 sigAlgebracsiga 34407 ×_s csx 34487 MblFnMcmbfm 34548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7401 df-oprab 7402 df-mpo 7403 df-1st 7972 df-2nd 7973 df-map 8812 df-siga 34408 df-sigagen 34438 df-sx 34488 df-mbfm 34549

This theorem is referenced by: (None)

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