Theorem ismbfm 32219

Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24792. (Contributed by Thierry Arnoux, 23-Jan-2017.)

Hypotheses

Ref	Expression
ismbfm.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
ismbfm.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)

Assertion

Ref	Expression
ismbfm	⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑇

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ismbfm

Dummy variables 𝑓 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismbfm.1	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2		ismbfm.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
3		unieq 4850	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
4	3	oveq2d 7291	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆))
5		eleq2 2827	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆))
6	5	ralbidv 3112	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆))
7	4, 6	rabeqbidv 3420	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆})
8		unieq 4850	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇)
9	8	oveq1d 7290	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆))
10		raleq 3342	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆))
11	9, 10	rabeqbidv 3420	. . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
12		df-mbfm 32218	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
13		ovex 7308	. . . . . 6 ⊢ (∪ 𝑇 ↑m ∪ 𝑆) ∈ V
14	13	rabex 5256	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V
15	7, 11, 12, 14	ovmpo 7433	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
16	1, 2, 15	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
17	16	eleq2d 2824	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}))
18		cnveq 5782	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
19	18	imaeq1d 5968	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥))
20	19	eleq1d 2823	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
21	20	ralbidv 3112	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
22	21	elrab 3624	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
23	17, 22	bitrdi 287	1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  ∪ cuni 4839  ^◡ccnv 5588  ran crn 5590   “ cima 5592  (class class class)co 7275   ↑_m cmap 8615  sigAlgebracsiga 32076  MblFnMcmbfm 32217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-mbfm 32218

This theorem is referenced by:  elunirnmbfm  32220  mbfmf  32222  isanmbfm  32223  mbfmcnvima  32224  mbfmcst  32226  1stmbfm  32227  2ndmbfm  32228  imambfm  32229  mbfmco  32231  elmbfmvol2  32234  mbfmcnt  32235  sibfof  32307  isrrvv  32410