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Theorem 1stmbfm 31417
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 7702 . . . 4 (1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 31351 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 584 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6493 . . . 4 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
71, 6mpbii 234 . . 3 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆)
8 unielsiga 31286 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
92, 8syl 17 . . . 4 (𝜑 𝑆𝑆)
10 sxsiga 31349 . . . . . 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 31286 . . . . 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 8409 . . 3 (𝜑 → ((1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)) ↔ (1st ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑆))
157, 14mpbird 258 . 2 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑆m (𝑆 ×s 𝑇)))
16 sgon 31282 . . . . . . . . . . 11 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
17 sigasspw 31274 . . . . . . . . . . 11 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
18 pwssb 5014 . . . . . . . . . . . 12 (𝑆 ⊆ 𝒫 𝑆 ↔ ∀𝑎𝑆 𝑎 𝑆)
1918biimpi 217 . . . . . . . . . . 11 (𝑆 ⊆ 𝒫 𝑆 → ∀𝑎𝑆 𝑎 𝑆)
202, 16, 17, 194syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑆 𝑎 𝑆)
2120r19.21bi 3205 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑎 𝑆)
22 xpss1 5567 . . . . . . . . 9 (𝑎 𝑆 → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ⊆ ( 𝑆 × 𝑇))
2423sseld 3963 . . . . . . 7 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 563 . . . . . 6 ((𝜑𝑎𝑆) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇))))
26 ffn 6507 . . . . . . . 8 ((1st ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑆 → (1st ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6820 . . . . . . . 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 6682 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) = (1st𝑧))
3029eleq1d 2894 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (1st𝑧) ∈ 𝑎))
31 1st2nd2 7717 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp2nd 7711 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (2nd𝑧) ∈ 𝑇)
33 elxp6 7712 . . . . . . . . . . . 12 (𝑧 ∈ (𝑎 × 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
34 anass 469 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑎 ∧ (2nd𝑧) ∈ 𝑇)))
35 an32 642 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑎) ∧ (2nd𝑧) ∈ 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
3633, 34, 353bitr2i 300 . . . . . . . . . . 11 (𝑧 ∈ (𝑎 × 𝑇) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) ∧ (1st𝑧) ∈ 𝑎))
3736baib 536 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
3831, 32, 37syl2anc 584 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ (𝑎 × 𝑇) ↔ (1st𝑧) ∈ 𝑎))
3930, 38bitr4d 283 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ (𝑎 × 𝑇)))
4039pm5.32i 575 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((1st ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
4128, 40bitri 276 . . . . . 6 (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ (𝑎 × 𝑇)))
4225, 41syl6rbbr 291 . . . . 5 ((𝜑𝑎𝑆) → (𝑧 ∈ ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ (𝑎 × 𝑇)))
4342eqrdv 2816 . . . 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 2818 . . . . . . . 8 𝑇 = 𝑇
48 issgon 31281 . . . . . . . 8 (𝑇 ∈ (sigAlgebra‘ 𝑇) ↔ (𝑇 ran sigAlgebra ∧ 𝑇 = 𝑇))
493, 47, 48sylanblrc 590 . . . . . . 7 (𝜑𝑇 ∈ (sigAlgebra‘ 𝑇))
50 baselsiga 31273 . . . . . . 7 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇𝑇)
5149, 50syl 17 . . . . . 6 (𝜑 𝑇𝑇)
5251adantr 481 . . . . 5 ((𝜑𝑎𝑆) → 𝑇𝑇)
53 elsx 31352 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ (𝑎𝑆 𝑇𝑇)) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5444, 45, 46, 52, 53syl22anc 834 . . . 4 ((𝜑𝑎𝑆) → (𝑎 × 𝑇) ∈ (𝑆 ×s 𝑇))
5543, 54eqeltrd 2910 . . 3 ((𝜑𝑎𝑆) → ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5655ralrimiva 3179 . 2 (𝜑 → ∀𝑎𝑆 ((1st ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5711, 2ismbfm 31409 . 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 709 1 (𝜑 → (1st ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  𝒫 cpw 4535  cop 4563   cuni 4830   × cxp 5546  ccnv 5547  ran crn 5549  cres 5550  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  m cmap 8395  sigAlgebracsiga 31266   ×s csx 31346  MblFnMcmbfm 31407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-siga 31267  df-sigagen 31297  df-sx 31347  df-mbfm 31408
This theorem is referenced by: (None)
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