Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  comptiunov2i Structured version   Visualization version   GIF version

Theorem comptiunov2i 38522
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 6069 . . 3 Fun 𝑋
3 comptiunov2.y . . . 4 𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))
43funmpt2 6069 . . 3 Fun 𝑌
5 funco 6070 . . 3 ((Fun 𝑋 ∧ Fun 𝑌) → Fun (𝑋𝑌))
62, 4, 5mp2an 672 . 2 Fun (𝑋𝑌)
7 comptiunov2.z . . 3 𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))
87funmpt2 6069 . 2 Fun 𝑍
9 ssv 3774 . . . . . . 7 ran 𝑌 ⊆ V
10 comptiunov2.i . . . . . . . . 9 𝐼 ∈ V
11 ovex 6827 . . . . . . . . 9 (𝑎 𝑖) ∈ V
1210, 11iunex 7298 . . . . . . . 8 𝑖𝐼 (𝑎 𝑖) ∈ V
1312, 1dmmpti 6162 . . . . . . 7 dom 𝑋 = V
149, 13sseqtr4i 3787 . . . . . 6 ran 𝑌 ⊆ dom 𝑋
15 dmcosseq 5524 . . . . . 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 6827 . . . . . . 7 (𝑏 𝑗) ∈ V
1917, 18iunex 7298 . . . . . 6 𝑗𝐽 (𝑏 𝑗) ∈ V
2019, 3dmmpti 6162 . . . . 5 dom 𝑌 = V
2116, 20eqtri 2793 . . . 4 dom (𝑋𝑌) = V
22 comptiunov2.k . . . . . . 7 𝐾 = (𝐼𝐽)
2310, 17unex 7107 . . . . . . 7 (𝐼𝐽) ∈ V
2422, 23eqeltri 2846 . . . . . 6 𝐾 ∈ V
25 ovex 6827 . . . . . 6 (𝑐 𝑘) ∈ V
2624, 25iunex 7298 . . . . 5 𝑘𝐾 (𝑐 𝑘) ∈ V
2726, 7dmmpti 6162 . . . 4 dom 𝑍 = V
2821, 27eqtr4i 2796 . . 3 dom (𝑋𝑌) = dom 𝑍
29 vex 3354 . . . . . . . . 9 𝑑 ∈ V
3029, 20eleqtrri 2849 . . . . . . . 8 𝑑 ∈ dom 𝑌
31 fvco 6418 . . . . . . . 8 ((Fun 𝑌𝑑 ∈ dom 𝑌) → ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑)))
324, 30, 31mp2an 672 . . . . . . 7 ((𝑋𝑌)‘𝑑) = (𝑋‘(𝑌𝑑))
33 oveq1 6803 . . . . . . . . . . 11 (𝑏 = 𝑑 → (𝑏 𝑗) = (𝑑 𝑗))
3433iuneq2d 4682 . . . . . . . . . 10 (𝑏 = 𝑑 𝑗𝐽 (𝑏 𝑗) = 𝑗𝐽 (𝑑 𝑗))
35 ovex 6827 . . . . . . . . . . 11 (𝑑 𝑗) ∈ V
3617, 35iunex 7298 . . . . . . . . . 10 𝑗𝐽 (𝑑 𝑗) ∈ V
3734, 3, 36fvmpt 6426 . . . . . . . . 9 (𝑑 ∈ V → (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗))
3829, 37ax-mp 5 . . . . . . . 8 (𝑌𝑑) = 𝑗𝐽 (𝑑 𝑗)
3938fveq2i 6336 . . . . . . 7 (𝑋‘(𝑌𝑑)) = (𝑋 𝑗𝐽 (𝑑 𝑗))
40 oveq1 6803 . . . . . . . . . 10 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → (𝑎 𝑖) = ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
4140iuneq2d 4682 . . . . . . . . 9 (𝑎 = 𝑗𝐽 (𝑑 𝑗) → 𝑖𝐼 (𝑎 𝑖) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖))
42 ovex 6827 . . . . . . . . . 10 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4310, 42iunex 7298 . . . . . . . . 9 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ∈ V
4441, 1, 43fvmpt 6426 . . . . . . . 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 2797 . . . . . 6 ((𝑋𝑌)‘𝑑) = 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
47 oveq1 6803 . . . . . . . . 9 (𝑐 = 𝑑 → (𝑐 𝑘) = (𝑑 𝑘))
4847iuneq2d 4682 . . . . . . . 8 (𝑐 = 𝑑 𝑘𝐾 (𝑐 𝑘) = 𝑘𝐾 (𝑑 𝑘))
49 ovex 6827 . . . . . . . . 9 (𝑑 𝑘) ∈ V
5024, 49iunex 7298 . . . . . . . 8 𝑘𝐾 (𝑑 𝑘) ∈ V
5148, 7, 50fvmpt 6426 . . . . . . 7 (𝑑 ∈ V → (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘))
5229, 51ax-mp 5 . . . . . 6 (𝑍𝑑) = 𝑘𝐾 (𝑑 𝑘)
5346, 52eqeq12i 2785 . . . . 5 (((𝑋𝑌)‘𝑑) = (𝑍𝑑) ↔ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
5421, 53raleqbii 3139 . . . 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 4740 . . . . . . . 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 3939 . . . . . . . 8 ( 𝑘𝐼 (𝑑 𝑘) ∪ 𝑘𝐽 (𝑑 𝑘)) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6056, 59eqsstri 3784 . . . . . . 7 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)
6155, 60eqssi 3768 . . . . . 6 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
62 iuneq1 4669 . . . . . . 7 (𝐾 = (𝐼𝐽) → 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘))
6322, 62ax-mp 5 . . . . . 6 𝑘𝐾 (𝑑 𝑘) = 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)
6461, 63eqtr4i 2796 . . . . 5 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘)
6564a1i 11 . . . 4 (𝑑 ∈ V → 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) = 𝑘𝐾 (𝑑 𝑘))
6654, 65mprgbir 3076 . . 3 𝑑 ∈ dom (𝑋𝑌)((𝑋𝑌)‘𝑑) = (𝑍𝑑)
67 eqfunfv 6461 . . . 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 678 . 2 ((Fun (𝑋𝑌) ∧ Fun 𝑍) → (𝑋𝑌) = 𝑍)
706, 8, 69mp2an 672 1 (𝑋𝑌) = 𝑍
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cun 3721  wss 3723   ciun 4655  cmpt 4864  dom cdm 5250  ran crn 5251  ccom 5254  Fun wfun 6024  cfv 6030  (class class class)co 6796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799
This theorem is referenced by:  corclrcl  38523  cotrcltrcl  38541  corcltrcl  38555  cotrclrcl  38558
  Copyright terms: Public domain W3C validator