Theorem 2ndmbfm 32861

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 7946	. . . 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 32792	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
5	2, 3, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
6	5	feq2d 6654	. . . 4 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
7	1, 6	mpbii 232	. . 3 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇)
8		unielsiga 32727	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
9	3, 8	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
10		sxsiga 32790	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
11	2, 3, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra)
12		unielsiga 32727	. . . . 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 8779	. . 3 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)) ↔ (2nd ↾ (∪ 𝑆 × ∪ 𝑇)):∪ (𝑆 ×s 𝑇)⟶∪ 𝑇))
15	7, 14	mpbird 256	. 2 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ (∪ 𝑇 ↑m ∪ (𝑆 ×s 𝑇)))
16		ffn 6668	. . . . . . . 8 ⊢ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇)):(∪ 𝑆 × ∪ 𝑇)⟶∪ 𝑇 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) Fn (∪ 𝑆 × ∪ 𝑇))
17		elpreima 7008	. . . . . . . 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 6861	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) = (2nd ‘𝑧))
20	19	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd ‘𝑧) ∈ 𝑎))
21		1st2nd2 7960	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
22		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (1st ‘𝑧) ∈ ∪ 𝑆)
23		elxp6 7955	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ ∪ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑎)))
24		anass 469	. . . . . . . . . . . 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 277	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) ∧ (2nd ‘𝑧) ∈ 𝑎))
26	25	baib 536	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ ∪ 𝑆) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
27	21, 22, 26	syl2anc 584	. . . . . . . . 9 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (2nd ‘𝑧) ∈ 𝑎))
28	20, 27	bitr4d 281	. . . . . . . 8 ⊢ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) → (((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎 ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
29	28	pm5.32i 575	. . . . . . 7 ⊢ ((𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ ((2nd ↾ (∪ 𝑆 × ∪ 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
30	18, 29	bitri 274	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
31		sgon 32723	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘∪ 𝑇))
32		sigasspw 32715	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (sigAlgebra‘∪ 𝑇) → 𝑇 ⊆ 𝒫 ∪ 𝑇)
33		pwssb 5061	. . . . . . . . . . . 12 ⊢ (𝑇 ⊆ 𝒫 ∪ 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ ∪ 𝑇)
34	33	biimpi 215	. . . . . . . . . . 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 3234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ ∪ 𝑇)
37		xpss2 5653	. . . . . . . . 9 ⊢ (𝑎 ⊆ ∪ 𝑇 → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
38	36, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (∪ 𝑆 × 𝑎) ⊆ (∪ 𝑆 × ∪ 𝑇))
39	38	sseld 3943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) → 𝑧 ∈ (∪ 𝑆 × ∪ 𝑇)))
40	39	pm4.71rd 563	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (∪ 𝑆 × 𝑎) ↔ (𝑧 ∈ (∪ 𝑆 × ∪ 𝑇) ∧ 𝑧 ∈ (∪ 𝑆 × 𝑎))))
41	30, 40	bitr4id 289	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑧 ∈ (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (∪ 𝑆 × 𝑎)))
42	41	eqrdv 2734	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) = (∪ 𝑆 × 𝑎))
43	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑆 ∈ ∪ ran sigAlgebra)
44	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑇 ∈ ∪ ran sigAlgebra)
45		eqid 2736	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
46		issgon 32722	. . . . . . . 8 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) ↔ (𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆))
47	2, 45, 46	sylanblrc 590	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝑆))
48		baselsiga 32714	. . . . . . 7 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → ∪ 𝑆 ∈ 𝑆)
49	47, 48	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
50	49	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∪ 𝑆 ∈ 𝑆)
51		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
52		elsx 32793	. . . . 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 2838	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
55	54	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 (◡(2nd ↾ (∪ 𝑆 × ∪ 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
56	11, 3	ismbfm 32850	. 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 711	1 ⊢ (𝜑 → (2nd ↾ (∪ 𝑆 × ∪ 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3910  𝒫 cpw 4560  ⟨cop 4592  ∪ cuni 4865   × cxp 5631  ^◡ccnv 5632  ran crn 5634   ↾ cres 5635   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920   ↑_m cmap 8765  sigAlgebracsiga 32707   ×_s csx 32787  MblFnMcmbfm 32848

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-siga 32708  df-sigagen 32738  df-sx 32788  df-mbfm 32849

This theorem is referenced by: (None)