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Theorem 2ndmbfm 30553
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 7310 . . . 4 (2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇
2 1stmbfm.1 . . . . . 6 (𝜑𝑆 ran sigAlgebra)
3 1stmbfm.2 . . . . . 6 (𝜑𝑇 ran sigAlgebra)
4 sxuni 30486 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
52, 3, 4syl2anc 696 . . . . 5 (𝜑 → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
65feq2d 6144 . . . 4 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
71, 6mpbii 223 . . 3 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇)
8 unielsiga 30421 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
93, 8syl 17 . . . 4 (𝜑 𝑇𝑇)
10 sxsiga 30484 . . . . . 6 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
112, 3, 10syl2anc 696 . . . . 5 (𝜑 → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
12 unielsiga 30421 . . . . 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 7988 . . 3 (𝜑 → ((2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇𝑚 (𝑆 ×s 𝑇)) ↔ (2nd ↾ ( 𝑆 × 𝑇)): (𝑆 ×s 𝑇)⟶ 𝑇))
157, 14mpbird 247 . 2 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ( 𝑇𝑚 (𝑆 ×s 𝑇)))
16 sgon 30417 . . . . . . . . . . 11 (𝑇 ran sigAlgebra → 𝑇 ∈ (sigAlgebra‘ 𝑇))
17 sigasspw 30409 . . . . . . . . . . 11 (𝑇 ∈ (sigAlgebra‘ 𝑇) → 𝑇 ⊆ 𝒫 𝑇)
18 pwssb 4720 . . . . . . . . . . . 12 (𝑇 ⊆ 𝒫 𝑇 ↔ ∀𝑎𝑇 𝑎 𝑇)
1918biimpi 206 . . . . . . . . . . 11 (𝑇 ⊆ 𝒫 𝑇 → ∀𝑎𝑇 𝑎 𝑇)
203, 16, 17, 194syl 19 . . . . . . . . . 10 (𝜑 → ∀𝑎𝑇 𝑎 𝑇)
2120r19.21bi 3034 . . . . . . . . 9 ((𝜑𝑎𝑇) → 𝑎 𝑇)
22 xpss2 5237 . . . . . . . . 9 (𝑎 𝑇 → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2321, 22syl 17 . . . . . . . 8 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ⊆ ( 𝑆 × 𝑇))
2423sseld 3708 . . . . . . 7 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) → 𝑧 ∈ ( 𝑆 × 𝑇)))
2524pm4.71rd 670 . . . . . 6 ((𝜑𝑎𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎))))
26 ffn 6158 . . . . . . . 8 ((2nd ↾ ( 𝑆 × 𝑇)):( 𝑆 × 𝑇)⟶ 𝑇 → (2nd ↾ ( 𝑆 × 𝑇)) Fn ( 𝑆 × 𝑇))
27 elpreima 6452 . . . . . . . 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 6320 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) = (2nd𝑧))
3029eleq1d 2788 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎 ↔ (2nd𝑧) ∈ 𝑎))
31 1st2nd2 7324 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32 xp1st 7317 . . . . . . . . . 10 (𝑧 ∈ ( 𝑆 × 𝑇) → (1st𝑧) ∈ 𝑆)
33 elxp6 7319 . . . . . . . . . . . 12 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
34 anass 684 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑎)))
3533, 34bitr4i 267 . . . . . . . . . . 11 (𝑧 ∈ ( 𝑆 × 𝑎) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) ∧ (2nd𝑧) ∈ 𝑎))
3635baib 982 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑆) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3731, 32, 36syl2anc 696 . . . . . . . . 9 (𝑧 ∈ ( 𝑆 × 𝑇) → (𝑧 ∈ ( 𝑆 × 𝑎) ↔ (2nd𝑧) ∈ 𝑎))
3830, 37bitr4d 271 . . . . . . . 8 (𝑧 ∈ ( 𝑆 × 𝑇) → (((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎𝑧 ∈ ( 𝑆 × 𝑎)))
3938pm5.32i 672 . . . . . . 7 ((𝑧 ∈ ( 𝑆 × 𝑇) ∧ ((2nd ↾ ( 𝑆 × 𝑇))‘𝑧) ∈ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4028, 39bitri 264 . . . . . 6 (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ (𝑧 ∈ ( 𝑆 × 𝑇) ∧ 𝑧 ∈ ( 𝑆 × 𝑎)))
4125, 40syl6rbbr 279 . . . . 5 ((𝜑𝑎𝑇) → (𝑧 ∈ ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ↔ 𝑧 ∈ ( 𝑆 × 𝑎)))
4241eqrdv 2722 . . . 4 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) = ( 𝑆 × 𝑎))
432adantr 472 . . . . 5 ((𝜑𝑎𝑇) → 𝑆 ran sigAlgebra)
443adantr 472 . . . . 5 ((𝜑𝑎𝑇) → 𝑇 ran sigAlgebra)
45 eqid 2724 . . . . . . . 8 𝑆 = 𝑆
46 issgon 30416 . . . . . . . 8 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
472, 45, 46sylanblrc 700 . . . . . . 7 (𝜑𝑆 ∈ (sigAlgebra‘ 𝑆))
48 baselsiga 30408 . . . . . . 7 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
4947, 48syl 17 . . . . . 6 (𝜑 𝑆𝑆)
5049adantr 472 . . . . 5 ((𝜑𝑎𝑇) → 𝑆𝑆)
51 simpr 479 . . . . 5 ((𝜑𝑎𝑇) → 𝑎𝑇)
52 elsx 30487 . . . . 5 (((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) ∧ ( 𝑆𝑆𝑎𝑇)) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5343, 44, 50, 51, 52syl22anc 1440 . . . 4 ((𝜑𝑎𝑇) → ( 𝑆 × 𝑎) ∈ (𝑆 ×s 𝑇))
5442, 53eqeltrd 2803 . . 3 ((𝜑𝑎𝑇) → ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5554ralrimiva 3068 . 2 (𝜑 → ∀𝑎𝑇 ((2nd ↾ ( 𝑆 × 𝑇)) “ 𝑎) ∈ (𝑆 ×s 𝑇))
5611, 3ismbfm 30544 . 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 995 1 (𝜑 → (2nd ↾ ( 𝑆 × 𝑇)) ∈ ((𝑆 ×s 𝑇)MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  wss 3680  𝒫 cpw 4266  cop 4291   cuni 4544   × cxp 5216  ccnv 5217  ran crn 5219  cres 5220  cima 5221   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  1st c1st 7283  2nd c2nd 7284  𝑚 cmap 7974  sigAlgebracsiga 30400   ×s csx 30481  MblFnMcmbfm 30542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976  df-siga 30401  df-sigagen 30432  df-sx 30482  df-mbfm 30543
This theorem is referenced by: (None)
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