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Theorem 1stmbfm 33722
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 8003 . . . 4 (1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 33654 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 583 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6703 . . . 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 33589 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
92, 8syl 17 . . . 4 (𝜑 𝑆𝑆)
10 sxsiga 33652 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 583 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 33589 . . . . 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 8840 . . 3 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)) ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
157, 14mpbird 257 . 2 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)))
16 ffn 6717 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 → (1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
17 elpreima 7059 . . . . . . . 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 6910 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) = (1st𝑧))
2019eleq1d 2817 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (1st𝑧) ∈ 𝑎))
21 1st2nd2 8018 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
22 xp2nd 8012 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (2nd𝑧) ∈ 𝑇)
23 elxp6 8013 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
24 anass 468 . . . . . . . . . . . 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 535 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
2821, 22, 27syl2anc 583 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
2920, 28bitr4d 282 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ (𝑎 × 𝑇)))
3029pm5.32i 574 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
3118, 30bitri 275 . . . . . 6 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
32 sgon 33585 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
33 sigasspw 33577 . . . . . . . . . . 11 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
34 pwssb 5104 . . . . . . . . . . . 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 3247 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑎 𝑆)
38 xpss1 5695 . . . . . . . . 9 (𝑎 𝑆 → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
3937, 38syl 17 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
4039sseld 3981 . . . . . . 7 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) → 𝑧 ∈ ( 𝑆 × 𝑇)))
4140pm4.71rd 562 . . . . . 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 2729 . . . 4 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) = (𝑎 × 𝑇))
442adantr 480 . . . . 5 ((𝜑𝑎𝑆) → 𝑆 ran sigAlgebra)
453adantr 480 . . . . 5 ((𝜑𝑎𝑆) → 𝑇 ran sigAlgebra)
46 simpr 484 . . . . 5 ((𝜑𝑎𝑆) → 𝑎𝑆)
47 eqid 2731 . . . . . . . 8 𝑇 = 𝑇
48 issgon 33584 . . . . . . . 8 (𝑇 ∈ (sigAlgebra‘ 𝑇) ↔ (𝑇 ran sigAlgebra ∧ 𝑇 = 𝑇))
493, 47, 48sylanblrc 589 . . . . . . 7 (𝜑𝑇 ∈ (sigAlgebra‘ 𝑇))
50 baselsiga 33576 . . . . . . 7 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇𝑇)
5149, 50syl 17 . . . . . 6 (𝜑 𝑇𝑇)
5251adantr 480 . . . . 5 ((𝜑𝑎𝑆) → 𝑇𝑇)
53 elsx 33655 . . . . 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 2832 . . 3 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 3145 . 2 (𝜑 → ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 2ismbfm 33712 . 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 395   = wceq 1540  wcel 2105  wral 3060  wss 3948  𝒫 cpw 4602  cop 4634   cuni 4908   × cxp 5674  ccnv 5675  ran crn 5677  cres 5678  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  1st c1st 7977  2nd c2nd 7978  m cmap 8826  sigAlgebracsiga 33569   ×s csx 33649  MblFnMcmbfm 33710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-siga 33570  df-sigagen 33600  df-sx 33650  df-mbfm 33711
This theorem is referenced by: (None)
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