Theorem mbfmcst 34770

Description: A constant function is measurable. Cf. mbfconst 25861. (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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑆) → 𝐴 ∈ ∪ 𝑇)
4	1, 3	fmpt3d 7109	. . 3 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
5		mbfmcst.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
6		unielsiga 34638	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
7	5, 6	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
8		mbfmcst.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
9		unielsiga 34638	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
11	7, 10	elmapd 8839	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ↔ 𝐹:∪ 𝑆⟶∪ 𝑇))
12	4, 11	mpbird 260	. 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆))
13		fconstmpt 5717	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × {𝐴}) = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
14	13	cnveqi 5854	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
15		cnvxp 6148	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ({𝐴} × ∪ 𝑆)
16	14, 15	eqtr3i 2785	. . . . . . . . 9 ⊢ ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) = ({𝐴} × ∪ 𝑆)
17	16	imaeq1i 6053	. . . . . . . 8 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = (({𝐴} × ∪ 𝑆) “ 𝑦)
18		df-ima 5668	. . . . . . . 8 ⊢ (({𝐴} × ∪ 𝑆) “ 𝑦) = ran (({𝐴} × ∪ 𝑆) ↾ 𝑦)
19		df-rn 5666	. . . . . . . 8 ⊢ ran (({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
20	17, 18, 19	3eqtri 2787	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
21		df-res 5667	. . . . . . . . . 10 ⊢ (({𝐴} × ∪ 𝑆) ↾ 𝑦) = (({𝐴} × ∪ 𝑆) ∩ (𝑦 × V))
22		inxp 5812	. . . . . . . . . 10 ⊢ (({𝐴} × ∪ 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × (∪ 𝑆 ∩ V))
23		inv1 4348	. . . . . . . . . . 11 ⊢ (∪ 𝑆 ∩ V) = ∪ 𝑆
24	23	xpeq2i 5682	. . . . . . . . . 10 ⊢ (({𝐴} ∩ 𝑦) × (∪ 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × ∪ 𝑆)
25	21, 22, 24	3eqtri 2787	. . . . . . . . 9 ⊢ (({𝐴} × ∪ 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × ∪ 𝑆)
26	25	cnveqi 5854	. . . . . . . 8 ⊢ ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
27	26	dmeqi 5888	. . . . . . 7 ⊢ dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
28		cnvxp 6148	. . . . . . . 8 ⊢ ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = (∪ 𝑆 × ({𝐴} ∩ 𝑦))
29	28	dmeqi 5888	. . . . . . 7 ⊢ dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
30	20, 27, 29	3eqtri 2787	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
31		xpeq2 5676	. . . . . . . . . . 11 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = (∪ 𝑆 × ∅))
32		xp0 5755	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × ∅) = ∅
33	31, 32	eqtrdi 2811	. . . . . . . . . 10 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
34	33	dmeqd 5889	. . . . . . . . 9 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35		dm0 5904	. . . . . . . . 9 ⊢ dom ∅ = ∅
36	34, 35	eqtrdi 2811	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
37	36	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38		0elsiga 34624	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
39	8, 38	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑆)
40	39	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
41	37, 40	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
42	30, 41	eqeltrid 2864	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
43		dmxp 5913	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) ≠ ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
44	43	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
45	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ∪ 𝑆 ∈ 𝑆)
46	44, 45	eqeltrd 2860	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
47	30, 46	eqeltrid 2864	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
48	42, 47	pm2.61dane 3042	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
49	48	ralrimivw 3158	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
50	1	cnveqd 5855	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴))
51	50	imaeq1d 6055	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦))
52	51	eleq1d 2845	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∈ 𝑆 ↔ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
53	52	ralbidv 3185	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
54	49, 53	mpbird 260	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)
55	8, 5	ismbfm 34762	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)))
56	12, 54, 55	mpbir2and 726	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ∩ cin 3898  ∅c0 4279  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  ⟶wf 6529  (class class class)co 7413   ↑_m cmap 8826  sigAlgebracsiga 34618  MblFnMcmbfm 34760

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8828  df-siga 34619  df-mbfm 34761

This theorem is used by:  sibf0  34845