Theorem comptiunov2i 44384

Description: The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)

Hypotheses

Ref	Expression
comptiunov2.x	⊢ 𝑋 = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ 𝐼 (𝑎 ↑ 𝑖))
comptiunov2.y	⊢ 𝑌 = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ 𝐽 (𝑏 ↑ 𝑗))
comptiunov2.z	⊢ 𝑍 = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ 𝐾 (𝑐 ↑ 𝑘))
comptiunov2.i	⊢ 𝐼 ∈ V
comptiunov2.j	⊢ 𝐽 ∈ V
comptiunov2.k	⊢ 𝐾 = (𝐼 ∪ 𝐽)
comptiunov2.1	⊢ ∪ 𝑘 ∈ 𝐼 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
comptiunov2.2	⊢ ∪ 𝑘 ∈ 𝐽 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
comptiunov2.3	⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) ⊆ ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘)

Assertion

Ref	Expression
comptiunov2i	⊢ (𝑋 ∘ 𝑌) = 𝑍

Distinct variable groups:   𝑖,𝑎, ↑   ↑ ,𝑏   ↑ ,𝑐   𝐼,𝑎,𝑖   𝑘,𝐼   𝑗,𝑎,𝐽,𝑖   𝐽,𝑏   𝑘,𝐽   𝑘,𝑐,𝐾   𝑋,𝑑   𝑌,𝑑   𝑍,𝑑   𝑎,𝑑,𝑖,𝑗   𝑏,𝑑,𝑗   𝑐,𝑑,𝑘

Allowed substitution hints:   ↑ (𝑗,𝑘,𝑑)   𝐼(𝑗,𝑏,𝑐,𝑑)   𝐽(𝑐,𝑑)   𝐾(𝑖,𝑗,𝑎,𝑏,𝑑)   𝑋(𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑍(𝑖,𝑗,𝑘,𝑎,𝑏,𝑐)

Proof of Theorem comptiunov2i

Step	Hyp	Ref	Expression
1		comptiunov2.x	. . . 4 ⊢ 𝑋 = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ 𝐼 (𝑎 ↑ 𝑖))
2	1	funmpt2 6579	. . 3 ⊢ Fun 𝑋
3		comptiunov2.y	. . . 4 ⊢ 𝑌 = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ 𝐽 (𝑏 ↑ 𝑗))
4	3	funmpt2 6579	. . 3 ⊢ Fun 𝑌
5		funco 6580	. . 3 ⊢ ((Fun 𝑋 ∧ Fun 𝑌) → Fun (𝑋 ∘ 𝑌))
6	2, 4, 5	mp2an 704	. 2 ⊢ Fun (𝑋 ∘ 𝑌)
7		comptiunov2.z	. . 3 ⊢ 𝑍 = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ 𝐾 (𝑐 ↑ 𝑘))
8	7	funmpt2 6579	. 2 ⊢ Fun 𝑍
9		ssv 3969	. . . . . . 7 ⊢ ran 𝑌 ⊆ V
10		comptiunov2.i	. . . . . . . . 9 ⊢ 𝐼 ∈ V
11		ovex 7447	. . . . . . . . 9 ⊢ (𝑎 ↑ 𝑖) ∈ V
12	10, 11	iunex 7968	. . . . . . . 8 ⊢ ∪ 𝑖 ∈ 𝐼 (𝑎 ↑ 𝑖) ∈ V
13	12, 1	dmmpti 6683	. . . . . . 7 ⊢ dom 𝑋 = V
14	9, 13	sseqtrri 3994	. . . . . 6 ⊢ ran 𝑌 ⊆ dom 𝑋
15		dmcosseq 5972	. . . . . 6 ⊢ (ran 𝑌 ⊆ dom 𝑋 → dom (𝑋 ∘ 𝑌) = dom 𝑌)
16	14, 15	ax-mp 5	. . . . 5 ⊢ dom (𝑋 ∘ 𝑌) = dom 𝑌
17		comptiunov2.j	. . . . . . 7 ⊢ 𝐽 ∈ V
18		ovex 7447	. . . . . . 7 ⊢ (𝑏 ↑ 𝑗) ∈ V
19	17, 18	iunex 7968	. . . . . 6 ⊢ ∪ 𝑗 ∈ 𝐽 (𝑏 ↑ 𝑗) ∈ V
20	19, 3	dmmpti 6683	. . . . 5 ⊢ dom 𝑌 = V
21	16, 20	eqtri 2793	. . . 4 ⊢ dom (𝑋 ∘ 𝑌) = V
22		comptiunov2.k	. . . . . . 7 ⊢ 𝐾 = (𝐼 ∪ 𝐽)
23	10, 17	unex 7746	. . . . . . 7 ⊢ (𝐼 ∪ 𝐽) ∈ V
24	22, 23	eqeltri 2866	. . . . . 6 ⊢ 𝐾 ∈ V
25		ovex 7447	. . . . . 6 ⊢ (𝑐 ↑ 𝑘) ∈ V
26	24, 25	iunex 7968	. . . . 5 ⊢ ∪ 𝑘 ∈ 𝐾 (𝑐 ↑ 𝑘) ∈ V
27	26, 7	dmmpti 6683	. . . 4 ⊢ dom 𝑍 = V
28	21, 27	eqtr4i 2796	. . 3 ⊢ dom (𝑋 ∘ 𝑌) = dom 𝑍
29		vex 3466	. . . . . . . . 9 ⊢ 𝑑 ∈ V
30	29, 20	eleqtrri 2869	. . . . . . . 8 ⊢ 𝑑 ∈ dom 𝑌
31		fvco 6983	. . . . . . . 8 ⊢ ((Fun 𝑌 ∧ 𝑑 ∈ dom 𝑌) → ((𝑋 ∘ 𝑌)‘𝑑) = (𝑋‘(𝑌‘𝑑)))
32	4, 30, 31	mp2an 704	. . . . . . 7 ⊢ ((𝑋 ∘ 𝑌)‘𝑑) = (𝑋‘(𝑌‘𝑑))
33		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (𝑏 ↑ 𝑗) = (𝑑 ↑ 𝑗))
34	33	iuneq2d 4992	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∪ 𝑗 ∈ 𝐽 (𝑏 ↑ 𝑗) = ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗))
35		ovex 7447	. . . . . . . . . . 11 ⊢ (𝑑 ↑ 𝑗) ∈ V
36	17, 35	iunex 7968	. . . . . . . . . 10 ⊢ ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ∈ V
37	34, 3, 36	fvmpt 6993	. . . . . . . . 9 ⊢ (𝑑 ∈ V → (𝑌‘𝑑) = ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗))
38	37	elv 3467	. . . . . . . 8 ⊢ (𝑌‘𝑑) = ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗)
39	38	fveq2i 6888	. . . . . . 7 ⊢ (𝑋‘(𝑌‘𝑑)) = (𝑋‘∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗))
40		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑎 = ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) → (𝑎 ↑ 𝑖) = (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖))
41	40	iuneq2d 4992	. . . . . . . . 9 ⊢ (𝑎 = ∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) → ∪ 𝑖 ∈ 𝐼 (𝑎 ↑ 𝑖) = ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖))
42		ovex 7447	. . . . . . . . . 10 ⊢ (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) ∈ V
43	10, 42	iunex 7968	. . . . . . . . 9 ⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) ∈ V
44	41, 1, 43	fvmpt 6993	. . . . . . . 8 ⊢ (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ∈ V → (𝑋‘∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗)) = ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖))
45	36, 44	ax-mp 5	. . . . . . 7 ⊢ (𝑋‘∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗)) = ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
46	32, 39, 45	3eqtri 2797	. . . . . 6 ⊢ ((𝑋 ∘ 𝑌)‘𝑑) = ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
47		oveq1 7421	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (𝑐 ↑ 𝑘) = (𝑑 ↑ 𝑘))
48	47	iuneq2d 4992	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ 𝐾 (𝑐 ↑ 𝑘) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘))
49		ovex 7447	. . . . . . . . 9 ⊢ (𝑑 ↑ 𝑘) ∈ V
50	24, 49	iunex 7968	. . . . . . . 8 ⊢ ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘) ∈ V
51	48, 7, 50	fvmpt 6993	. . . . . . 7 ⊢ (𝑑 ∈ V → (𝑍‘𝑑) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘))
52	51	elv 3467	. . . . . 6 ⊢ (𝑍‘𝑑) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘)
53	46, 52	eqeq12i 2788	. . . . 5 ⊢ (((𝑋 ∘ 𝑌)‘𝑑) = (𝑍‘𝑑) ↔ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘))
54	21, 53	raleqbii 3343	. . . 4 ⊢ (∀𝑑 ∈ dom (𝑋 ∘ 𝑌)((𝑋 ∘ 𝑌)‘𝑑) = (𝑍‘𝑑) ↔ ∀𝑑 ∈ V ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘))
55		comptiunov2.3	. . . . . . 7 ⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) ⊆ ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘)
56		iunxun 5065	. . . . . . . 8 ⊢ ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘) = (∪ 𝑘 ∈ 𝐼 (𝑑 ↑ 𝑘) ∪ ∪ 𝑘 ∈ 𝐽 (𝑑 ↑ 𝑘))
57		comptiunov2.1	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ 𝐼 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
58		comptiunov2.2	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ 𝐽 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
59	57, 58	unssi 4152	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ 𝐼 (𝑑 ↑ 𝑘) ∪ ∪ 𝑘 ∈ 𝐽 (𝑑 ↑ 𝑘)) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
60	56, 59	eqsstri 3991	. . . . . . 7 ⊢ ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)
61	55, 60	eqssi 3961	. . . . . 6 ⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) = ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘)
62		iuneq1 4978	. . . . . . 7 ⊢ (𝐾 = (𝐼 ∪ 𝐽) → ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘) = ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘))
63	22, 62	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘) = ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘)
64	61, 63	eqtr4i 2796	. . . . 5 ⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘)
65	64	a1i 11	. . . 4 ⊢ (𝑑 ∈ V → ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) = ∪ 𝑘 ∈ 𝐾 (𝑑 ↑ 𝑘))
66	54, 65	mprgbir 3093	. . 3 ⊢ ∀𝑑 ∈ dom (𝑋 ∘ 𝑌)((𝑋 ∘ 𝑌)‘𝑑) = (𝑍‘𝑑)
67		eqfunfv 7035	. . . 4 ⊢ ((Fun (𝑋 ∘ 𝑌) ∧ Fun 𝑍) → ((𝑋 ∘ 𝑌) = 𝑍 ↔ (dom (𝑋 ∘ 𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋 ∘ 𝑌)((𝑋 ∘ 𝑌)‘𝑑) = (𝑍‘𝑑))))
68	67	biimprd 251	. . 3 ⊢ ((Fun (𝑋 ∘ 𝑌) ∧ Fun 𝑍) → ((dom (𝑋 ∘ 𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋 ∘ 𝑌)((𝑋 ∘ 𝑌)‘𝑑) = (𝑍‘𝑑)) → (𝑋 ∘ 𝑌) = 𝑍))
69	28, 66, 68	mp2ani 710	. 2 ⊢ ((Fun (𝑋 ∘ 𝑌) ∧ Fun 𝑍) → (𝑋 ∘ 𝑌) = 𝑍)
70	6, 8, 69	mp2an 704	1 ⊢ (𝑋 ∘ 𝑌) = 𝑍

Syntax hints:   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   ∪ cun 3911   ⊆ wss 3913  ∪ ciun 4961   ↦ cmpt 5197  dom cdm 5665  ran crn 5666   ∘ ccom 5669  Fun wfun 6534  ‘cfv 6540  (class class class)co 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-ov 7417

This theorem is referenced by:  corclrcl  44385  cotrcltrcl  44403  corcltrcl  44417  cotrclrcl  44420