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Theorem comptiunov2i 37479
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 5885 . . 3 Fun 𝑋
3 comptiunov2.y . . . 4 𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))
43funmpt2 5885 . . 3 Fun 𝑌
5 funco 5886 . . 3 ((Fun 𝑋 ∧ Fun 𝑌) → Fun (𝑋𝑌))
62, 4, 5mp2an 707 . 2 Fun (𝑋𝑌)
7 comptiunov2.z . . 3 𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))
87funmpt2 5885 . 2 Fun 𝑍
9 ssv 3604 . . . . . . 7 ran 𝑌 ⊆ V
10 comptiunov2.i . . . . . . . . 9 𝐼 ∈ V
11 ovex 6632 . . . . . . . . 9 (𝑎 𝑖) ∈ V
1210, 11iunex 7093 . . . . . . . 8 𝑖𝐼 (𝑎 𝑖) ∈ V
1312, 1dmmpti 5980 . . . . . . 7 dom 𝑋 = V
149, 13sseqtr4i 3617 . . . . . 6 ran 𝑌 ⊆ dom 𝑋
15 dmcosseq 5347 . . . . . 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 6632 . . . . . . 7 (𝑏 𝑗) ∈ V
1917, 18iunex 7093 . . . . . 6 𝑗𝐽 (𝑏 𝑗) ∈ V
2019, 3dmmpti 5980 . . . . 5 dom 𝑌 = V
2116, 20eqtri 2643 . . . 4 dom (𝑋𝑌) = V
22 comptiunov2.k . . . . . . 7 𝐾 = (𝐼𝐽)
2310, 17unex 6909 . . . . . . 7 (𝐼𝐽) ∈ V
2422, 23eqeltri 2694 . . . . . 6 𝐾 ∈ V
25 ovex 6632 . . . . . 6 (𝑐 𝑘) ∈ V
2624, 25iunex 7093 . . . . 5 𝑘𝐾 (𝑐 𝑘) ∈ V
2726, 7dmmpti 5980 . . . 4 dom 𝑍 = V
2821, 27eqtr4i 2646 . . 3 dom (𝑋𝑌) = dom 𝑍
29 vex 3189 . . . . . . . . 9 𝑑 ∈ V
3029, 20eleqtrri 2697 . . . . . . . 8 𝑑 ∈ dom 𝑌
31 fvco 6231 . . . . . . . 8 ((Fun 𝑌𝑑 ∈ dom 𝑌) → ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑)))
324, 30, 31mp2an 707 . . . . . . 7 ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑))
33 oveq1 6611 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏 𝑗) = (𝑑 𝑗))
3433iuneq2d 4513 . . . . . . . . . 10 (𝑏 = 𝑑 𝑗𝐽 (𝑏 𝑗) = 𝑗𝐽 (𝑑 𝑗))
35 ovex 6632 . . . . . . . . . . 11 (𝑑 𝑗) ∈ V
3617, 35iunex 7093 . . . . . . . . . 10 𝑗𝐽 (𝑑 𝑗) ∈ V
3734, 3, 36fvmpt 6239 . . . . . . . . 9 (𝑑 ∈ V → (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗))
3829, 37ax-mp 5 . . . . . . . 8 (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗)
3938fveq2i 6151 . . . . . . 7 (𝑋‘(𝑌𝑑)) = (𝑋 𝑗𝐽 (𝑑 𝑗))
40 oveq1 6611 . . . . . . . . . 10 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → (𝑎 𝑖) = ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
4140iuneq2d 4513 . . . . . . . . 9 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → 𝑖𝐼 (𝑎 𝑖) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
42 ovex 6632 . . . . . . . . . 10 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4310, 42iunex 7093 . . . . . . . . 9 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4441, 1, 43fvmpt 6239 . . . . . . . 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 2647 . . . . . 6 ((𝑋𝑌)‘𝑑) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
47 oveq1 6611 . . . . . . . . 9 (𝑐 = 𝑑 → (𝑐 𝑘) = (𝑑 𝑘))
4847iuneq2d 4513 . . . . . . . 8 (𝑐 = 𝑑 𝑘𝐾 (𝑐 𝑘) = 𝑘𝐾 (𝑑 𝑘))
49 ovex 6632 . . . . . . . . 9 (𝑑 𝑘) ∈ V
5024, 49iunex 7093 . . . . . . . 8 𝑘𝐾 (𝑑 𝑘) ∈ V
5148, 7, 50fvmpt 6239 . . . . . . 7 (𝑑 ∈ V → (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘))
5229, 51ax-mp 5 . . . . . 6 (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘)
5346, 52eqeq12i 2635 . . . . 5 (((𝑋𝑌)‘𝑑) = (𝑍𝑑) ↔ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
5421, 53raleqbii 2984 . . . 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 4571 . . . . . . . 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 3766 . . . . . . . 8 ( 𝑘𝐼 (𝑑 𝑘) ∪ 𝑘𝐽 (𝑑 𝑘)) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6056, 59eqsstri 3614 . . . . . . 7 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6155, 60eqssi 3599 . . . . . 6 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
62 iuneq1 4500 . . . . . . 7 (𝐾 = (𝐼𝐽) → 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘))
6322, 62ax-mp 5 . . . . . 6 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
6461, 63eqtr4i 2646 . . . . 5 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘)
6564a1i 11 . . . 4 (𝑑 ∈ V → 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
6654, 65mprgbir 2922 . . 3 𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)
67 eqfunfv 6272 . . . 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 713 . 2 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → (𝑋𝑌) = 𝑍)
706, 8, 69mp2an 707 1 (𝑋𝑌) = 𝑍
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cun 3553  wss 3555   ciun 4485  cmpt 4673  dom cdm 5074  ran crn 5075  ccom 5078  Fun wfun 5841  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607
This theorem is referenced by:  corclrcl  37480  cotrcltrcl  37498  corcltrcl  37512  cotrclrcl  37515
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