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Theorem 2ndmbfm 34243
Description: The second 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
2ndmbfm (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))

Proof of Theorem 2ndmbfm
Dummy variables 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f2ndres 8038 . . . 4 (2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 34174 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 584 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6723 . . . 4 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
71, 6mpbii 233 . . 3 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇)
8 unielsiga 34109 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
93, 8syl 17 . . . 4 (𝜑 𝑇𝑇)
10 sxsiga 34172 . . . . . 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 34109 . . . . 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 8879 . . 3 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)) ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
157, 14mpbird 257 . 2 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)))
16 ffn 6737 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 → (2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
17 elpreima 7078 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
181, 16, 17mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
19 fvres 6926 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) = (2nd𝑧))
2019eleq1d 2824 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd𝑧) ∈ 𝑎))
21 1st2nd2 8052 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
22 xp1st 8045 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (1st𝑧) ∈ 𝑆)
23 elxp6 8047 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
24 anass 468 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
2523, 24bitr4i 278 . . . . . . . . . . 11 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎))
2625baib 535 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
2721, 22, 26syl2anc 584 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
2820, 27bitr4d 282 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ ( 𝑆 × 𝑎)))
2928pm5.32i 574 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
3018, 29bitri 275 . . . . . 6 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
31 sgon 34105 . . . . . . . . . . 11 (𝑇 ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘ 𝑇))
32 sigasspw 34097 . . . . . . . . . . 11 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇 ⊆ 𝒫 𝑇)
33 pwssb 5106 . . . . . . . . . . . 12 (𝑇 ⊆ 𝒫 𝑇 ↔ ∀𝑎𝑇 𝑎 𝑇)
3433biimpi 216 . . . . . . . . . . 11 (𝑇 ⊆ 𝒫 𝑇 → ∀𝑎𝑇 𝑎 𝑇)
353, 31, 32, 344syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑇 𝑎 𝑇)
3635r19.21bi 3249 . . . . . . . . 9 ((𝜑𝑎𝑇) → 𝑎 𝑇)
37 xpss2 5709 . . . . . . . . 9 (𝑎 𝑇 → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
3836, 37syl 17 . . . . . . . 8 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
3938sseld 3994 . . . . . . 7 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) → 𝑧 ∈ ( 𝑆 × 𝑇)))
4039pm4.71rd 562 . . . . . 6 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎))))
4130, 40bitr4id 290 . . . . 5 ((𝜑𝑎𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ ( 𝑆 × 𝑎)))
4241eqrdv 2733 . . . 4 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) = ( 𝑆 × 𝑎))
432adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
443adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
45 eqid 2735 . . . . . . . 8 𝑆 = 𝑆
46 issgon 34104 . . . . . . . 8 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
472, 45, 46sylanblrc 590 . . . . . . 7 (𝜑𝑆 ∈ (sigAlgebra‘ 𝑆))
48 baselsiga 34096 . . . . . . 7 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
4947, 48syl 17 . . . . . 6 (𝜑 𝑆𝑆)
5049adantr 480 . . . . 5 ((𝜑𝑎𝑇) → 𝑆𝑆)
51 simpr 484 . . . . 5 ((𝜑𝑎𝑇) → 𝑎𝑇)
52 elsx 34175 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ ( 𝑆𝑆𝑎𝑇)) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5343, 44, 50, 51, 52syl22anc 839 . . . 4 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5442, 53eqeltrd 2839 . . 3 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5554ralrimiva 3144 . 2 (𝜑 → ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5611, 3ismbfm 34232 . 2 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇) ↔ ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)) ∧ ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))))
5715, 55, 56mpbir2and 713 1 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  𝒫 cpw 4605  cop 4637   cuni 4912   × cxp 5687  ccnv 5688  ran crn 5690  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  sigAlgebracsiga 34089   ×s csx 34169  MblFnMcmbfm 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-siga 34090  df-sigagen 34120  df-sx 34170  df-mbfm 34231
This theorem is referenced by: (None)
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