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Theorem 1stmbfm 32227
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 7855 . . . 4 (1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 32161 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 584 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6586 . . . 4 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
71, 6mpbii 232 . . 3 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆)
8 unielsiga 32096 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
92, 8syl 17 . . . 4 (𝜑 𝑆𝑆)
10 sxsiga 32159 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 584 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 32096 . . . . 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 8629 . . 3 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)) ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
157, 14mpbird 256 . 2 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)))
16 ffn 6600 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 → (1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
17 elpreima 6935 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
181, 16, 17mp2b 10 . . . . . . 7 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
19 fvres 6793 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) = (1st𝑧))
2019eleq1d 2823 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (1st𝑧) ∈ 𝑎))
21 1st2nd2 7870 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
22 xp2nd 7864 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (2nd𝑧) ∈ 𝑇)
23 elxp6 7865 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
24 anass 469 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
25 an32 643 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
2623, 24, 253bitr2i 299 . . . . . . . . . . 11 (𝑧 ∈ (𝑎 × 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
2726baib 536 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
2821, 22, 27syl2anc 584 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
2920, 28bitr4d 281 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ (𝑎 × 𝑇)))
3029pm5.32i 575 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
3118, 30bitri 274 . . . . . 6 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
32 sgon 32092 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
33 sigasspw 32084 . . . . . . . . . . 11 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
34 pwssb 5030 . . . . . . . . . . . 12 (𝑆 ⊆ 𝒫 𝑆 ↔ ∀𝑎𝑆 𝑎 𝑆)
3534biimpi 215 . . . . . . . . . . 11 (𝑆 ⊆ 𝒫 𝑆 → ∀𝑎𝑆 𝑎 𝑆)
362, 32, 33, 354syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑆 𝑎 𝑆)
3736r19.21bi 3134 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑎 𝑆)
38 xpss1 5608 . . . . . . . . 9 (𝑎 𝑆 → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
3937, 38syl 17 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
4039sseld 3920 . . . . . . 7 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) → 𝑧 ∈ ( 𝑆 × 𝑇)))
4140pm4.71rd 563 . . . . . 6 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇))))
4231, 41bitr4id 290 . . . . 5 ((𝜑𝑎𝑆) → (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (𝑎 × 𝑇)))
4342eqrdv 2736 . . . 4 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) = (𝑎 × 𝑇))
442adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑆 ran sigAlgebra)
453adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑇 ran sigAlgebra)
46 simpr 485 . . . . 5 ((𝜑𝑎𝑆) → 𝑎𝑆)
47 eqid 2738 . . . . . . . 8 𝑇 = 𝑇
48 issgon 32091 . . . . . . . 8 (𝑇 ∈ (sigAlgebra‘ 𝑇) ↔ (𝑇 ran sigAlgebra ∧ 𝑇 = 𝑇))
493, 47, 48sylanblrc 590 . . . . . . 7 (𝜑𝑇 ∈ (sigAlgebra‘ 𝑇))
50 baselsiga 32083 . . . . . . 7 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇𝑇)
5149, 50syl 17 . . . . . 6 (𝜑 𝑇𝑇)
5251adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑇𝑇)
53 elsx 32162 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ (𝑎𝑆 𝑇𝑇)) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5444, 45, 46, 52, 53syl22anc 836 . . . 4 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5543, 54eqeltrd 2839 . . 3 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 3103 . 2 (𝜑 → ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 2ismbfm 32219 . 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 710 1 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wss 3887  𝒫 cpw 4533  cop 4567   cuni 4839   × cxp 5587  ccnv 5588  ran crn 5590  cres 5591  cima 5592   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  m cmap 8615  sigAlgebracsiga 32076   ×s csx 32156  MblFnMcmbfm 32217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-siga 32077  df-sigagen 32107  df-sx 32157  df-mbfm 32218
This theorem is referenced by: (None)
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