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Theorem 1stmbfm 30103
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 (𝜑𝑆 ran sigAlgebra)
1stmbfm.2 (𝜑𝑇 ran sigAlgebra)
Assertion
Ref Expression
1stmbfm (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))

Proof of Theorem 1stmbfm
Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 7135 . . . 4 (1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 30037 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 692 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 5988 . . . 4 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
71, 6mpbii 223 . . 3 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆)
8 unielsiga 29972 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
92, 8syl 17 . . . 4 (𝜑 𝑆𝑆)
10 sxsiga 30035 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 692 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 29972 . . . . 5 ((𝑆 ×s 𝑇) ∈ ran sigAlgebra → (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
1311, 12syl 17 . . . 4 (𝜑 (𝑆 ×s 𝑇) ∈ (𝑆 ×s 𝑇))
149, 13elmapd 7816 . . 3 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆𝑚 (𝑆 ×s 𝑇)) ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
157, 14mpbird 247 . 2 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆𝑚 (𝑆 ×s 𝑇)))
16 sgon 29968 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
17 sigasspw 29960 . . . . . . . . . . 11 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
18 pwssb 4578 . . . . . . . . . . . 12 (𝑆 ⊆ 𝒫 𝑆 ↔ ∀𝑎𝑆 𝑎 𝑆)
1918biimpi 206 . . . . . . . . . . 11 (𝑆 ⊆ 𝒫 𝑆 → ∀𝑎𝑆 𝑎 𝑆)
202, 16, 17, 194syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑆 𝑎 𝑆)
2120r19.21bi 2927 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑎 𝑆)
22 xpss1 5189 . . . . . . . . 9 (𝑎 𝑆 → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2423sseld 3582 . . . . . . 7 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 666 . . . . . 6 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇))))
26 ffn 6002 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 → (1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6293 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
281, 26, 27mp2b 10 . . . . . . 7 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
29 fvres 6164 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) = (1st𝑧))
3029eleq1d 2683 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (1st𝑧) ∈ 𝑎))
31 1st2nd2 7150 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp2nd 7144 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (2nd𝑧) ∈ 𝑇)
33 elxp6 7145 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
34 anass 680 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
35 an32 838 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
3633, 34, 353bitr2i 288 . . . . . . . . . . 11 (𝑧 ∈ (𝑎 × 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
3736baib 943 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
3831, 32, 37syl2anc 692 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
3930, 38bitr4d 271 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ (𝑎 × 𝑇)))
4039pm5.32i 668 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
4128, 40bitri 264 . . . . . 6 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
4225, 41syl6rbbr 279 . . . . 5 ((𝜑𝑎𝑆) → (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (𝑎 × 𝑇)))
4342eqrdv 2619 . . . 4 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) = (𝑎 × 𝑇))
442adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑆 ran sigAlgebra)
453adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑇 ran sigAlgebra)
46 simpr 477 . . . . 5 ((𝜑𝑎𝑆) → 𝑎𝑆)
47 eqid 2621 . . . . . . . 8 𝑇 = 𝑇
48 issgon 29967 . . . . . . . 8 (𝑇 ∈ (sigAlgebra‘ 𝑇) ↔ (𝑇 ran sigAlgebra ∧ 𝑇 = 𝑇))
493, 47, 48sylanblrc 696 . . . . . . 7 (𝜑𝑇 ∈ (sigAlgebra‘ 𝑇))
50 baselsiga 29959 . . . . . . 7 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇𝑇)
5149, 50syl 17 . . . . . 6 (𝜑 𝑇𝑇)
5251adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑇𝑇)
53 elsx 30038 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ (𝑎𝑆 𝑇𝑇)) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5444, 45, 46, 52, 53syl22anc 1324 . . . 4 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5543, 54eqeltrd 2698 . . 3 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 2960 . 2 (𝜑 → ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 2ismbfm 30095 . 2 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆) ↔ ((1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆𝑚 (𝑆 ×s 𝑇)) ∧ ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
5815, 56, 57mpbir2and 956 1 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555  𝒫 cpw 4130  cop 4154   cuni 4402   × cxp 5072  ccnv 5073  ran crn 5075  cres 5076  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  𝑚 cmap 7802  sigAlgebracsiga 29951   ×s csx 30032  MblFnMcmbfm 30093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-siga 29952  df-sigagen 29983  df-sx 30033  df-mbfm 30094
This theorem is referenced by: (None)
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