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Theorem comptiunov2i 40844
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 ↦ 𝑖𝐼 (𝑎 𝑖))
comptiunov2.y 𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))
comptiunov2.z 𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))
comptiunov2.i 𝐼 ∈ V
comptiunov2.j 𝐽 ∈ V
comptiunov2.k 𝐾 = (𝐼𝐽)
comptiunov2.1 𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
comptiunov2.2 𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
comptiunov2.3 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
Assertion
Ref Expression
comptiunov2i (𝑋𝑌) = 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
StepHypRef Expression
1 comptiunov2.x . . . 4 𝑋 = (𝑎 ∈ V ↦ 𝑖𝐼 (𝑎 𝑖))
21funmpt2 6372 . . 3 Fun 𝑋
3 comptiunov2.y . . . 4 𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))
43funmpt2 6372 . . 3 Fun 𝑌
5 funco 6373 . . 3 ((Fun 𝑋 ∧ Fun 𝑌) → Fun (𝑋𝑌))
62, 4, 5mp2an 692 . 2 Fun (𝑋𝑌)
7 comptiunov2.z . . 3 𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))
87funmpt2 6372 . 2 Fun 𝑍
9 ssv 3899 . . . . . . 7 ran 𝑌 ⊆ V
10 comptiunov2.i . . . . . . . . 9 𝐼 ∈ V
11 ovex 7197 . . . . . . . . 9 (𝑎 𝑖) ∈ V
1210, 11iunex 7687 . . . . . . . 8 𝑖𝐼 (𝑎 𝑖) ∈ V
1312, 1dmmpti 6475 . . . . . . 7 dom 𝑋 = V
149, 13sseqtrri 3912 . . . . . 6 ran 𝑌 ⊆ dom 𝑋
15 dmcosseq 5810 . . . . . 6 (ran 𝑌 ⊆ dom 𝑋 → dom (𝑋𝑌) = dom 𝑌)
1614, 15ax-mp 5 . . . . 5 dom (𝑋𝑌) = dom 𝑌
17 comptiunov2.j . . . . . . 7 𝐽 ∈ V
18 ovex 7197 . . . . . . 7 (𝑏 𝑗) ∈ V
1917, 18iunex 7687 . . . . . 6 𝑗𝐽 (𝑏 𝑗) ∈ V
2019, 3dmmpti 6475 . . . . 5 dom 𝑌 = V
2116, 20eqtri 2761 . . . 4 dom (𝑋𝑌) = V
22 comptiunov2.k . . . . . . 7 𝐾 = (𝐼𝐽)
2310, 17unex 7481 . . . . . . 7 (𝐼𝐽) ∈ V
2422, 23eqeltri 2829 . . . . . 6 𝐾 ∈ V
25 ovex 7197 . . . . . 6 (𝑐 𝑘) ∈ V
2624, 25iunex 7687 . . . . 5 𝑘𝐾 (𝑐 𝑘) ∈ V
2726, 7dmmpti 6475 . . . 4 dom 𝑍 = V
2821, 27eqtr4i 2764 . . 3 dom (𝑋𝑌) = dom 𝑍
29 vex 3401 . . . . . . . . 9 𝑑 ∈ V
3029, 20eleqtrri 2832 . . . . . . . 8 𝑑 ∈ dom 𝑌
31 fvco 6760 . . . . . . . 8 ((Fun 𝑌𝑑 ∈ dom 𝑌) → ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑)))
324, 30, 31mp2an 692 . . . . . . 7 ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑))
33 oveq1 7171 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏 𝑗) = (𝑑 𝑗))
3433iuneq2d 4907 . . . . . . . . . 10 (𝑏 = 𝑑 𝑗𝐽 (𝑏 𝑗) = 𝑗𝐽 (𝑑 𝑗))
35 ovex 7197 . . . . . . . . . . 11 (𝑑 𝑗) ∈ V
3617, 35iunex 7687 . . . . . . . . . 10 𝑗𝐽 (𝑑 𝑗) ∈ V
3734, 3, 36fvmpt 6769 . . . . . . . . 9 (𝑑 ∈ V → (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗))
3837elv 3403 . . . . . . . 8 (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗)
3938fveq2i 6671 . . . . . . 7 (𝑋‘(𝑌𝑑)) = (𝑋 𝑗𝐽 (𝑑 𝑗))
40 oveq1 7171 . . . . . . . . . 10 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → (𝑎 𝑖) = ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
4140iuneq2d 4907 . . . . . . . . 9 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → 𝑖𝐼 (𝑎 𝑖) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
42 ovex 7197 . . . . . . . . . 10 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4310, 42iunex 7687 . . . . . . . . 9 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4441, 1, 43fvmpt 6769 . . . . . . . 8 ( 𝑗𝐽 (𝑑 𝑗) ∈ V → (𝑋 𝑗𝐽 (𝑑 𝑗)) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
4536, 44ax-mp 5 . . . . . . 7 (𝑋 𝑗𝐽 (𝑑 𝑗)) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
4632, 39, 453eqtri 2765 . . . . . 6 ((𝑋𝑌)‘𝑑) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
47 oveq1 7171 . . . . . . . . 9 (𝑐 = 𝑑 → (𝑐 𝑘) = (𝑑 𝑘))
4847iuneq2d 4907 . . . . . . . 8 (𝑐 = 𝑑 𝑘𝐾 (𝑐 𝑘) = 𝑘𝐾 (𝑑 𝑘))
49 ovex 7197 . . . . . . . . 9 (𝑑 𝑘) ∈ V
5024, 49iunex 7687 . . . . . . . 8 𝑘𝐾 (𝑑 𝑘) ∈ V
5148, 7, 50fvmpt 6769 . . . . . . 7 (𝑑 ∈ V → (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘))
5251elv 3403 . . . . . 6 (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘)
5346, 52eqeq12i 2753 . . . . 5 (((𝑋𝑌)‘𝑑) = (𝑍𝑑) ↔ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
5421, 53raleqbii 3146 . . . 4 (∀𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑) ↔ ∀𝑑 ∈ V 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
55 comptiunov2.3 . . . . . . 7 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
56 iunxun 4976 . . . . . . . 8 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘) = ( 𝑘𝐼 (𝑑 𝑘) ∪ 𝑘𝐽 (𝑑 𝑘))
57 comptiunov2.1 . . . . . . . . 9 𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
58 comptiunov2.2 . . . . . . . . 9 𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
5957, 58unssi 4073 . . . . . . . 8 ( 𝑘𝐼 (𝑑 𝑘) ∪ 𝑘𝐽 (𝑑 𝑘)) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6056, 59eqsstri 3909 . . . . . . 7 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6155, 60eqssi 3891 . . . . . 6 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
62 iuneq1 4894 . . . . . . 7 (𝐾 = (𝐼𝐽) → 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘))
6322, 62ax-mp 5 . . . . . 6 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
6461, 63eqtr4i 2764 . . . . 5 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘)
6564a1i 11 . . . 4 (𝑑 ∈ V → 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
6654, 65mprgbir 3068 . . 3 𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)
67 eqfunfv 6808 . . . 4 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → ((𝑋𝑌) = 𝑍 ↔ (dom (𝑋𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑))))
6867biimprd 251 . . 3 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → ((dom (𝑋𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)) → (𝑋𝑌) = 𝑍))
6928, 66, 68mp2ani 698 . 2 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → (𝑋𝑌) = 𝑍)
706, 8, 69mp2an 692 1 (𝑋𝑌) = 𝑍
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2113  wral 3053  Vcvv 3397  cun 3839  wss 3841   ciun 4878  cmpt 5107  dom cdm 5519  ran crn 5520  ccom 5523  Fun wfun 6327  cfv 6333  (class class class)co 7164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167
This theorem is referenced by:  corclrcl  40845  cotrcltrcl  40863  corcltrcl  40877  cotrclrcl  40880
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