Theorem ismbfm 32119

Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24697. (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 4847	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
4	3	oveq2d 7271	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆))
5		eleq2 2827	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆))
6	5	ralbidv 3120	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆))
7	4, 6	rabeqbidv 3410	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆})
8		unieq 4847	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇)
9	8	oveq1d 7270	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆))
10		raleq 3333	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆))
11	9, 10	rabeqbidv 3410	. . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
12		df-mbfm 32118	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
13		ovex 7288	. . . . . 6 ⊢ (∪ 𝑇 ↑m ∪ 𝑆) ∈ V
14	13	rabex 5251	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V
15	7, 11, 12, 14	ovmpo 7411	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
16	1, 2, 15	syl2anc 583	. . 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 5771	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
19	18	imaeq1d 5957	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥))
20	19	eleq1d 2823	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
21	20	ralbidv 3120	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
22	21	elrab 3617	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
23	17, 22	bitrdi 286	1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  ∪ cuni 4836  ^◡ccnv 5579  ran crn 5581   “ cima 5583  (class class class)co 7255   ↑_m cmap 8573  sigAlgebracsiga 31976  MblFnMcmbfm 32117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-mbfm 32118

This theorem is referenced by:  elunirnmbfm  32120  mbfmf  32122  isanmbfm  32123  mbfmcnvima  32124  mbfmcst  32126  1stmbfm  32127  2ndmbfm  32128  imambfm  32129  mbfmco  32131  elmbfmvol2  32134  mbfmcnt  32135  sibfof  32207  isrrvv  32310