Theorem mbfmcst 34261

Description: A constant function is measurable. Cf. mbfconst 25668. (Contributed by Thierry Arnoux, 26-Jan-2017.)

Hypotheses

Ref	Expression
mbfmcst.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
mbfmcst.2	⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
mbfmcst.3	⊢ (𝜑 → 𝐹 = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴))
mbfmcst.4	⊢ (𝜑 → 𝐴 ∈ ∪ 𝑇)

Assertion

Ref	Expression
mbfmcst	⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆   𝑥,𝑇   𝜑,𝑥

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem mbfmcst

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmcst.3	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴))
2		mbfmcst.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑇)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑆) → 𝐴 ∈ ∪ 𝑇)
4	1, 3	fmpt3d 7136	. . 3 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
5		mbfmcst.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
6		unielsiga 34129	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
8		mbfmcst.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
9		unielsiga 34129	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
11	7, 10	elmapd 8880	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ↔ 𝐹:∪ 𝑆⟶∪ 𝑇))
12	4, 11	mpbird 257	. 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆))
13		fconstmpt 5747	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × {𝐴}) = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
14	13	cnveqi 5885	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
15		cnvxp 6177	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ({𝐴} × ∪ 𝑆)
16	14, 15	eqtr3i 2767	. . . . . . . . 9 ⊢ ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) = ({𝐴} × ∪ 𝑆)
17	16	imaeq1i 6075	. . . . . . . 8 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = (({𝐴} × ∪ 𝑆) “ 𝑦)
18		df-ima 5698	. . . . . . . 8 ⊢ (({𝐴} × ∪ 𝑆) “ 𝑦) = ran (({𝐴} × ∪ 𝑆) ↾ 𝑦)
19		df-rn 5696	. . . . . . . 8 ⊢ ran (({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
20	17, 18, 19	3eqtri 2769	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
21		df-res 5697	. . . . . . . . . 10 ⊢ (({𝐴} × ∪ 𝑆) ↾ 𝑦) = (({𝐴} × ∪ 𝑆) ∩ (𝑦 × V))
22		inxp 5842	. . . . . . . . . 10 ⊢ (({𝐴} × ∪ 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × (∪ 𝑆 ∩ V))
23		inv1 4398	. . . . . . . . . . 11 ⊢ (∪ 𝑆 ∩ V) = ∪ 𝑆
24	23	xpeq2i 5712	. . . . . . . . . 10 ⊢ (({𝐴} ∩ 𝑦) × (∪ 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × ∪ 𝑆)
25	21, 22, 24	3eqtri 2769	. . . . . . . . 9 ⊢ (({𝐴} × ∪ 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × ∪ 𝑆)
26	25	cnveqi 5885	. . . . . . . 8 ⊢ ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
27	26	dmeqi 5915	. . . . . . 7 ⊢ dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
28		cnvxp 6177	. . . . . . . 8 ⊢ ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = (∪ 𝑆 × ({𝐴} ∩ 𝑦))
29	28	dmeqi 5915	. . . . . . 7 ⊢ dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
30	20, 27, 29	3eqtri 2769	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
31		xpeq2 5706	. . . . . . . . . . 11 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = (∪ 𝑆 × ∅))
32		xp0 6178	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × ∅) = ∅
33	31, 32	eqtrdi 2793	. . . . . . . . . 10 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
34	33	dmeqd 5916	. . . . . . . . 9 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35		dm0 5931	. . . . . . . . 9 ⊢ dom ∅ = ∅
36	34, 35	eqtrdi 2793	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
37	36	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38		0elsiga 34115	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
39	8, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑆)
40	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
41	37, 40	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
42	30, 41	eqeltrid 2845	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
43		dmxp 5939	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) ≠ ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
44	43	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
45	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ∪ 𝑆 ∈ 𝑆)
46	44, 45	eqeltrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
47	30, 46	eqeltrid 2845	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
48	42, 47	pm2.61dane 3029	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
49	48	ralrimivw 3150	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
50	1	cnveqd 5886	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴))
51	50	imaeq1d 6077	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦))
52	51	eleq1d 2826	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∈ 𝑆 ↔ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
53	52	ralbidv 3178	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
54	49, 53	mpbird 257	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)
55	8, 5	ismbfm 34252	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)))
56	12, 54, 55	mpbir2and 713	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   ∩ cin 3950  ∅c0 4333  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688  ⟶wf 6557  (class class class)co 7431   ↑_m cmap 8866  sigAlgebracsiga 34109  MblFnMcmbfm 34250

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-siga 34110  df-mbfm 34251

This theorem is referenced by:  sibf0  34336