Theorem 2ndmbfm 31629

Description: The second 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
2ndmbfm	⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Proof of Theorem 2ndmbfm

Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f2ndres 7696	. . . 4 ⊢ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇
2		1stmbfm.1	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
3		1stmbfm.2	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
4		sxuni 31562	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
5	2, 3, 4	syl2anc 587	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
6	5	feq2d 6473	. . . 4 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
7	1, 6	mpbii 236	. . 3 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇)
8		unielsiga 31497	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
9	3, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
10		sxsiga 31560	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 31497	. . . . 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 8403	. . 3 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)) ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
15	7, 14	mpbird 260	. 2 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)))
16		ffn 6487	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
17		elpreima 6805	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎)))
18	1, 16, 17	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎))
19		fvres 6664	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (2nd ‘𝑧))
20	19	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd ‘𝑧) ∈ 𝑎))
21		1st2nd2 7710	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
22		xp1st 7703	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (1st ‘𝑧) ∈ ∪ 𝑆)
23		elxp6 7705	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
24		anass 472	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
25	23, 24	bitr4i 281	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎))
26	25	baib 539	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
27	21, 22, 26	syl2anc 587	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
28	20, 27	bitr4d 285	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
29	28	pm5.32i 578	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
30	18, 29	bitri 278	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
31		sgon 31493	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
32		sigasspw 31485	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → 𝑇 ⊆ 𝒫 ∪ 𝑇)
33		pwssb 4986	. . . . . . . . . . . 12 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
34	33	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
35	3, 31, 32, 34	4syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
36	35	r19.21bi 3173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ ∪ 𝑇)
37		xpss2 5539	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑇 → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
38	36, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
39	38	sseld 3914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
40	39	pm4.71rd 566	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎))))
41	30, 40	bitr4id 293	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
42	41	eqrdv 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (∪ 𝑆 × 𝑎))
43	2	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
44	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
45		eqid 2798	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
46		issgon 31492	. . . . . . . 8 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆))
47	2, 45, 46	sylanblrc 593	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
48		baselsiga 31484	. . . . . . 7 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → ∪ 𝑆 ∈ 𝑆)
49	47, 48	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
50	49	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∪ 𝑆 ∈ 𝑆)
51		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
52		elsx 31563	. . . . 5 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) ∧ (∪ 𝑆 ∈ 𝑆 ∧ 𝑎 ∈ 𝑇)) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
53	43, 44, 50, 51, 52	syl22anc 837	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
54	42, 53	eqeltrd 2890	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
55	54	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
56	11, 3	ismbfm 31620	. 2 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇) ↔ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)) ∧ ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
57	15, 55, 56	mpbir2and 712	1 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  𝒫 cpw 4497  ⟨cop 4531  ∪ cuni 4800   × cxp 5517  ^◡ccnv 5518  ran crn 5520   ↾ cres 5521   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670   ↑_m cmap 8389  sigAlgebracsiga 31477   ×_s csx 31557  MblFnMcmbfm 31618

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-siga 31478  df-sigagen 31508  df-sx 31558  df-mbfm 31619

This theorem is referenced by: (None)