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Theorem 1stmbfm 30631
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 7357 . . . 4 (1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 30565 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 696 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6192 . . . 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 30500 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
92, 8syl 17 . . . 4 (𝜑 𝑆𝑆)
10 sxsiga 30563 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 696 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 30500 . . . . 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 8037 . . 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 30496 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
17 sigasspw 30488 . . . . . . . . . . 11 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
18 pwssb 4764 . . . . . . . . . . . 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 3070 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑎 𝑆)
22 xpss1 5284 . . . . . . . . 9 (𝑎 𝑆 → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2423sseld 3743 . . . . . . 7 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 670 . . . . . 6 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇))))
26 ffn 6206 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 → (1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6500 . . . . . . . 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 6368 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) = (1st𝑧))
3029eleq1d 2824 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (1st𝑧) ∈ 𝑎))
31 1st2nd2 7372 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp2nd 7366 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (2nd𝑧) ∈ 𝑇)
33 elxp6 7367 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
34 anass 684 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
35 an32 874 . . . . . . . . . . . 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 982 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
3831, 32, 37syl2anc 696 . . . . . . . . 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 672 . . . . . . 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 2758 . . . 4 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) = (𝑎 × 𝑇))
442adantr 472 . . . . 5 ((𝜑𝑎𝑆) → 𝑆 ran sigAlgebra)
453adantr 472 . . . . 5 ((𝜑𝑎𝑆) → 𝑇 ran sigAlgebra)
46 simpr 479 . . . . 5 ((𝜑𝑎𝑆) → 𝑎𝑆)
47 eqid 2760 . . . . . . . 8 𝑇 = 𝑇
48 issgon 30495 . . . . . . . 8 (𝑇 ∈ (sigAlgebra‘ 𝑇) ↔ (𝑇 ran sigAlgebra ∧ 𝑇 = 𝑇))
493, 47, 48sylanblrc 700 . . . . . . 7 (𝜑𝑇 ∈ (sigAlgebra‘ 𝑇))
50 baselsiga 30487 . . . . . . 7 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇𝑇)
5149, 50syl 17 . . . . . 6 (𝜑 𝑇𝑇)
5251adantr 472 . . . . 5 ((𝜑𝑎𝑆) → 𝑇𝑇)
53 elsx 30566 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ (𝑎𝑆 𝑇𝑇)) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5444, 45, 46, 52, 53syl22anc 1478 . . . 4 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5543, 54eqeltrd 2839 . . 3 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 3104 . 2 (𝜑 → ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 2ismbfm 30623 . 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 995 1 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wss 3715  𝒫 cpw 4302  cop 4327   cuni 4588   × cxp 5264  ccnv 5265  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  sigAlgebracsiga 30479   ×s csx 30560  MblFnMcmbfm 30621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-siga 30480  df-sigagen 30511  df-sx 30561  df-mbfm 30622
This theorem is referenced by: (None)
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