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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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