Theorem ismbfm 34650

Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 25798. (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 4882	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆)
4	3	oveq2d 7428	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆))
5		eleq2 2851	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆))
6	5	ralbidv 3187	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆))
7	4, 6	rabeqbidv 3433	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆})
8		unieq 4882	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇)
9	8	oveq1d 7427	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆))
10		raleq 3319	. . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆))
11	9, 10	rabeqbidv 3433	. . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
12		df-mbfm 34649	. . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠})
13		ovex 7445	. . . . . 6 ⊢ (∪ 𝑇 ↑m ∪ 𝑆) ∈ V
14	13	rabex 5308	. . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V
15	7, 11, 12, 14	ovmpo 7572	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
16	1, 2, 15	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})
17	16	eleq2d 2848	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}))
18		cnveq 5858	. . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
19	18	imaeq1d 6060	. . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥))
20	19	eleq1d 2847	. . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
21	20	ralbidv 3187	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
22	21	elrab 3649	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))
23	17, 22	bitrdi 290	1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415  ∪ cuni 4871  ^◡ccnv 5659  ran crn 5661   “ cima 5663  (class class class)co 7412   ↑_m cmap 8822  sigAlgebracsiga 34507  MblFnMcmbfm 34648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-mbfm 34649

This theorem is used by:  elunirnmbfm  34651  mbfmf  34653  mbfmcnvima  34654  mbfmcst  34658  1stmbfm  34659  2ndmbfm  34660  imambfm  34661  mbfmco  34663  elmbfmvol2  34666  mbfmcnt  34667  sibfof  34739  isrrvv  34842