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Theorem 2ndmbfm 31539
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 7699 . . . 4 (2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 31472 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 587 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6483 . . . 4 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
71, 6mpbii 236 . . 3 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇)
8 unielsiga 31407 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
93, 8syl 17 . . . 4 (𝜑 𝑇𝑇)
10 sxsiga 31470 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 587 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 31407 . . . . 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 8405 . . 3 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)) ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
157, 14mpbird 260 . 2 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇m (𝑆 ×s 𝑇)))
16 sgon 31403 . . . . . . . . . . 11 (𝑇 ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘ 𝑇))
17 sigasspw 31395 . . . . . . . . . . 11 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇 ⊆ 𝒫 𝑇)
18 pwssb 5006 . . . . . . . . . . . 12 (𝑇 ⊆ 𝒫 𝑇 ↔ ∀𝑎𝑇 𝑎 𝑇)
1918biimpi 219 . . . . . . . . . . 11 (𝑇 ⊆ 𝒫 𝑇 → ∀𝑎𝑇 𝑎 𝑇)
203, 16, 17, 194syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑇 𝑎 𝑇)
2120r19.21bi 3202 . . . . . . . . 9 ((𝜑𝑎𝑇) → 𝑎 𝑇)
22 xpss2 5558 . . . . . . . . 9 (𝑎 𝑇 → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2423sseld 3950 . . . . . . 7 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 566 . . . . . 6 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎))))
26 ffn 6497 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 → (2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6811 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎)))
281, 26, 27mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎))
29 fvres 6672 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) = (2nd𝑧))
3029eleq1d 2900 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd𝑧) ∈ 𝑎))
31 1st2nd2 7713 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp1st 7706 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (1st𝑧) ∈ 𝑆)
33 elxp6 7708 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
34 anass 472 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
3533, 34bitr4i 281 . . . . . . . . . . 11 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎))
3635baib 539 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3731, 32, 36syl2anc 587 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3830, 37bitr4d 285 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ ( 𝑆 × 𝑎)))
3938pm5.32i 578 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4028, 39bitri 278 . . . . . 6 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4125, 40syl6rbbr 293 . . . . 5 ((𝜑𝑎𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ ( 𝑆 × 𝑎)))
4241eqrdv 2822 . . . 4 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) = ( 𝑆 × 𝑎))
432adantr 484 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
443adantr 484 . . . . 5 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
45 eqid 2824 . . . . . . . 8 𝑆 = 𝑆
46 issgon 31402 . . . . . . . 8 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
472, 45, 46sylanblrc 593 . . . . . . 7 (𝜑𝑆 ∈ (sigAlgebra‘ 𝑆))
48 baselsiga 31394 . . . . . . 7 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
4947, 48syl 17 . . . . . 6 (𝜑 𝑆𝑆)
5049adantr 484 . . . . 5 ((𝜑𝑎𝑇) → 𝑆𝑆)
51 simpr 488 . . . . 5 ((𝜑𝑎𝑇) → 𝑎𝑇)
52 elsx 31473 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ ( 𝑆𝑆𝑎𝑇)) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5343, 44, 50, 51, 52syl22anc 837 . . . 4 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5442, 53eqeltrd 2916 . . 3 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5554ralrimiva 3176 . 2 (𝜑 → ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5611, 3ismbfm 31530 . 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 712 1 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  wss 3918  𝒫 cpw 4520  cop 4554   cuni 4821   × cxp 5536  ccnv 5537  ran crn 5539  cres 5540  cima 5541   Fn wfn 6333  wf 6334  cfv 6338  (class class class)co 7140  1st c1st 7672  2nd c2nd 7673  m cmap 8391  sigAlgebracsiga 31387   ×s csx 31467  MblFnMcmbfm 31528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-map 8393  df-siga 31388  df-sigagen 31418  df-sx 31468  df-mbfm 31529
This theorem is referenced by: (None)
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