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Theorem comptiunov2i 38315
 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 5965 . . 3 Fun 𝑋
3 comptiunov2.y . . . 4 𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))
43funmpt2 5965 . . 3 Fun 𝑌
5 funco 5966 . . 3 ((Fun 𝑋 ∧ Fun 𝑌) → Fun (𝑋𝑌))
62, 4, 5mp2an 708 . 2 Fun (𝑋𝑌)
7 comptiunov2.z . . 3 𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))
87funmpt2 5965 . 2 Fun 𝑍
9 ssv 3658 . . . . . . 7 ran 𝑌 ⊆ V
10 comptiunov2.i . . . . . . . . 9 𝐼 ∈ V
11 ovex 6718 . . . . . . . . 9 (𝑎 𝑖) ∈ V
1210, 11iunex 7189 . . . . . . . 8 𝑖𝐼 (𝑎 𝑖) ∈ V
1312, 1dmmpti 6061 . . . . . . 7 dom 𝑋 = V
149, 13sseqtr4i 3671 . . . . . 6 ran 𝑌 ⊆ dom 𝑋
15 dmcosseq 5419 . . . . . 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 6718 . . . . . . 7 (𝑏 𝑗) ∈ V
1917, 18iunex 7189 . . . . . 6 𝑗𝐽 (𝑏 𝑗) ∈ V
2019, 3dmmpti 6061 . . . . 5 dom 𝑌 = V
2116, 20eqtri 2673 . . . 4 dom (𝑋𝑌) = V
22 comptiunov2.k . . . . . . 7 𝐾 = (𝐼𝐽)
2310, 17unex 6998 . . . . . . 7 (𝐼𝐽) ∈ V
2422, 23eqeltri 2726 . . . . . 6 𝐾 ∈ V
25 ovex 6718 . . . . . 6 (𝑐 𝑘) ∈ V
2624, 25iunex 7189 . . . . 5 𝑘𝐾 (𝑐 𝑘) ∈ V
2726, 7dmmpti 6061 . . . 4 dom 𝑍 = V
2821, 27eqtr4i 2676 . . 3 dom (𝑋𝑌) = dom 𝑍
29 vex 3234 . . . . . . . . 9 𝑑 ∈ V
3029, 20eleqtrri 2729 . . . . . . . 8 𝑑 ∈ dom 𝑌
31 fvco 6313 . . . . . . . 8 ((Fun 𝑌𝑑 ∈ dom 𝑌) → ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑)))
324, 30, 31mp2an 708 . . . . . . 7 ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑))
33 oveq1 6697 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏 𝑗) = (𝑑 𝑗))
3433iuneq2d 4579 . . . . . . . . . 10 (𝑏 = 𝑑 𝑗𝐽 (𝑏 𝑗) = 𝑗𝐽 (𝑑 𝑗))
35 ovex 6718 . . . . . . . . . . 11 (𝑑 𝑗) ∈ V
3617, 35iunex 7189 . . . . . . . . . 10 𝑗𝐽 (𝑑 𝑗) ∈ V
3734, 3, 36fvmpt 6321 . . . . . . . . 9 (𝑑 ∈ V → (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗))
3829, 37ax-mp 5 . . . . . . . 8 (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗)
3938fveq2i 6232 . . . . . . 7 (𝑋‘(𝑌𝑑)) = (𝑋 𝑗𝐽 (𝑑 𝑗))
40 oveq1 6697 . . . . . . . . . 10 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → (𝑎 𝑖) = ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
4140iuneq2d 4579 . . . . . . . . 9 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → 𝑖𝐼 (𝑎 𝑖) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
42 ovex 6718 . . . . . . . . . 10 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4310, 42iunex 7189 . . . . . . . . 9 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4441, 1, 43fvmpt 6321 . . . . . . . 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 2677 . . . . . 6 ((𝑋𝑌)‘𝑑) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
47 oveq1 6697 . . . . . . . . 9 (𝑐 = 𝑑 → (𝑐 𝑘) = (𝑑 𝑘))
4847iuneq2d 4579 . . . . . . . 8 (𝑐 = 𝑑 𝑘𝐾 (𝑐 𝑘) = 𝑘𝐾 (𝑑 𝑘))
49 ovex 6718 . . . . . . . . 9 (𝑑 𝑘) ∈ V
5024, 49iunex 7189 . . . . . . . 8 𝑘𝐾 (𝑑 𝑘) ∈ V
5148, 7, 50fvmpt 6321 . . . . . . 7 (𝑑 ∈ V → (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘))
5229, 51ax-mp 5 . . . . . 6 (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘)
5346, 52eqeq12i 2665 . . . . 5 (((𝑋𝑌)‘𝑑) = (𝑍𝑑) ↔ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
5421, 53raleqbii 3019 . . . 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 4637 . . . . . . . 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 3821 . . . . . . . 8 ( 𝑘𝐼 (𝑑 𝑘) ∪ 𝑘𝐽 (𝑑 𝑘)) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6056, 59eqsstri 3668 . . . . . . 7 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6155, 60eqssi 3652 . . . . . 6 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
62 iuneq1 4566 . . . . . . 7 (𝐾 = (𝐼𝐽) → 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘))
6322, 62ax-mp 5 . . . . . 6 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
6461, 63eqtr4i 2676 . . . . 5 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘)
6564a1i 11 . . . 4 (𝑑 ∈ V → 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
6654, 65mprgbir 2956 . . 3 𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)
67 eqfunfv 6356 . . . 4 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → ((𝑋𝑌) = 𝑍 ↔ (dom (𝑋𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑))))
6867biimprd 238 . . 3 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → ((dom (𝑋𝑌) = dom 𝑍 ∧ ∀𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)) → (𝑋𝑌) = 𝑍))
6928, 66, 68mp2ani 714 . 2 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → (𝑋𝑌) = 𝑍)
706, 8, 69mp2an 708 1 (𝑋𝑌) = 𝑍
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ∪ ciun 4552   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   ∘ ccom 5147  Fun wfun 5920  ‘cfv 5926  (class class class)co 6690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693 This theorem is referenced by:  corclrcl  38316  cotrcltrcl  38334  corcltrcl  38348  cotrclrcl  38351
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