Theorem ismbfm 34224

Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 25527. (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 4869	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
4	3	oveq2d 7365	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆))
5		eleq2 2817	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆))
6	5	ralbidv 3152	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆))
7	4, 6	rabeqbidv 3413	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆})
8		unieq 4869	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇)
9	8	oveq1d 7364	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆))
10		raleq 3286	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆))
11	9, 10	rabeqbidv 3413	. . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
12		df-mbfm 34223	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
13		ovex 7382	. . . . . 6 ⊢ (∪ 𝑇 ↑m ∪ 𝑆) ∈ V
14	13	rabex 5278	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V
15	7, 11, 12, 14	ovmpo 7509	. . . 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 2814	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}))
18		cnveq 5816	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
19	18	imaeq1d 6010	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥))
20	19	eleq1d 2813	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
21	20	ralbidv 3152	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
22	21	elrab 3648	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  ∪ cuni 4858  ^◡ccnv 5618  ran crn 5620   “ cima 5622  (class class class)co 7349   ↑_m cmap 8753  sigAlgebracsiga 34081  MblFnMcmbfm 34222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-mbfm 34223

This theorem is referenced by:  elunirnmbfm  34225  mbfmf  34227  mbfmcnvima  34229  mbfmcst  34233  1stmbfm  34234  2ndmbfm  34235  imambfm  34236  mbfmco  34238  elmbfmvol2  34241  mbfmcnt  34242  sibfof  34314  isrrvv  34417