Theorem mbfmcst 35002

Description: A constant function is measurable. Cf. mbfconst 26009. (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 7148	. . 3 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇)
5		mbfmcst.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
6		unielsiga 34871	. . . . 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 34871	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
11	7, 10	elmapd 8898	. . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ↔ 𝐹:∪ 𝑆⟶∪ 𝑇))
12	4, 11	mpbird 260	. 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆))
13		fconstmpt 5720	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × {𝐴}) = (𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
14	13	cnveqi 5864	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴)
15		cnvxp 6165	. . . . . . . . . 10 ⊢ ◡(∪ 𝑆 × {𝐴}) = ({𝐴} × ∪ 𝑆)
16	14, 15	eqtr3i 2788	. . . . . . . . 9 ⊢ ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) = ({𝐴} × ∪ 𝑆)
17	16	imaeq1i 6067	. . . . . . . 8 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = (({𝐴} × ∪ 𝑆) “ 𝑦)
18		df-ima 5671	. . . . . . . 8 ⊢ (({𝐴} × ∪ 𝑆) “ 𝑦) = ran (({𝐴} × ∪ 𝑆) ↾ 𝑦)
19		df-rn 5669	. . . . . . . 8 ⊢ ran (({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
20	17, 18, 19	3eqtri 2790	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦)
21		xpres 6024	. . . . . . . . 9 ⊢ (({𝐴} × ∪ 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × ∪ 𝑆)
22	21	cnveqi 5864	. . . . . . . 8 ⊢ ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
23	22	dmeqi 5898	. . . . . . 7 ⊢ dom ◡(({𝐴} × ∪ 𝑆) ↾ 𝑦) = dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆)
24		cnvxp 6165	. . . . . . . 8 ⊢ ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = (∪ 𝑆 × ({𝐴} ∩ 𝑦))
25	24	dmeqi 5898	. . . . . . 7 ⊢ dom ◡(({𝐴} ∩ 𝑦) × ∪ 𝑆) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
26	20, 23, 25	3eqtri 2790	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) = dom (∪ 𝑆 × ({𝐴} ∩ 𝑦))
27		xpeq2 5679	. . . . . . . . . . 11 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = (∪ 𝑆 × ∅))
28		xp0 5759	. . . . . . . . . . 11 ⊢ (∪ 𝑆 × ∅) = ∅
29	27, 28	eqtrdi 2814	. . . . . . . . . 10 ⊢ (({𝐴} ∩ 𝑦) = ∅ → (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
30	29	dmeqd 5899	. . . . . . . . 9 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
31		dm0 5914	. . . . . . . . 9 ⊢ dom ∅ = ∅
32	30, 31	eqtrdi 2814	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) = ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
33	32	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
34		0elsiga 34857	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
35	8, 34	syl 18	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑆)
36	35	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
37	33, 36	eqeltrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
38	26, 37	eqeltrid 2868	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
39		dmxp 5923	. . . . . . . 8 ⊢ (({𝐴} ∩ 𝑦) ≠ ∅ → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
40	39	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) = ∪ 𝑆)
41	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ∪ 𝑆 ∈ 𝑆)
42	40, 41	eqeltrd 2864	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom (∪ 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
43	26, 42	eqeltrid 2868	. . . . 5 ⊢ ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
44	38, 43	pm2.61dane 3046	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
45	44	ralrimivw 3162	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆)
46	1	cnveqd 5865	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴))
47	46	imaeq1d 6069	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦))
48	47	eleq1d 2849	. . . 4 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∈ 𝑆 ↔ (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
49	48	ralbidv 3189	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑇 (◡(𝑥 ∈ ∪ 𝑆 ↦ 𝐴) “ 𝑦) ∈ 𝑆))
50	45, 49	mpbird 260	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)
51	8, 5	ismbfm 34995	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑦 ∈ 𝑇 (◡𝐹 “ 𝑦) ∈ 𝑆)))
52	12, 50, 51	mpbir2and 726	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ∩ cin 4092  ∅c0 4281  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192   × cxp 5656  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   ↾ cres 5660   “ cima 5661  ⟶wf 6562  (class class class)co 7451   ↑_m cmap 8885  sigAlgebracsiga 34851  MblFnMcmbfm 34993

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-siga 34852  df-mbfm 34994

This theorem is used by:  sibf0  35077