Theorem isanmbfm 32223

Description: The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
mbfmf.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmf.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmf.3	⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

Assertion

Ref	Expression
isanmbfm	⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Proof of Theorem isanmbfm

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

Step	Hyp	Ref	Expression
1		mbfmf.1	. . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2		mbfmf.2	. . 3 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
3		mbfmf.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))
4	1, 2	ismbfm 32219	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))
5	3, 4	mpbid 231	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
6		unieq 4850	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
7	6	oveq2d 7291	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆))
8	7	eleq2d 2824	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ↔ 𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆)))
9		eleq2 2827	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((◡𝐹 “ 𝑥) ∈ 𝑠 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
10	9	ralbidv 3112	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆))
11	8, 10	anbi12d 631	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆)))
12		unieq 4850	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇)
13	12	oveq1d 7290	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆))
14	13	eleq2d 2824	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ↔ 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆)))
15		raleq 3342	. . . . 5 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
16	14, 15	anbi12d 631	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))
17	11, 16	rspc2ev 3572	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
18	1, 2, 5, 17	syl3anc 1370	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
19		elunirnmbfm 32220	. 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))
20	18, 19	sylibr 233	1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ∪ 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  ax-un 7588

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-csb 3833  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-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  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-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-mbfm 32218

This theorem is referenced by:  mbfmbfm  32225  orvcval4  32427