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Theorem ituniiun 9282
Description: Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)
Hypothesis
Ref Expression
ituni.u 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
Assertion
Ref Expression
ituniiun (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑎   𝑥,𝐵,𝑦,𝑎   𝑈,𝑎
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑎)

Proof of Theorem ituniiun
Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . 4 (𝑏 = 𝐴 → (𝑈𝑏) = (𝑈𝐴))
21fveq1d 6231 . . 3 (𝑏 = 𝐴 → ((𝑈𝑏)‘suc 𝐵) = ((𝑈𝐴)‘suc 𝐵))
3 iuneq1 4566 . . 3 (𝑏 = 𝐴 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
42, 3eqeq12d 2666 . 2 (𝑏 = 𝐴 → (((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵) ↔ ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵)))
5 suceq 5828 . . . . . 6 (𝑑 = ∅ → suc 𝑑 = suc ∅)
65fveq2d 6233 . . . . 5 (𝑑 = ∅ → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc ∅))
7 fveq2 6229 . . . . . 6 (𝑑 = ∅ → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘∅))
87iuneq2d 4579 . . . . 5 (𝑑 = ∅ → 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘∅))
96, 8eqeq12d 2666 . . . 4 (𝑑 = ∅ → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)))
10 suceq 5828 . . . . . 6 (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
1110fveq2d 6233 . . . . 5 (𝑑 = 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝑐))
12 fveq2 6229 . . . . . 6 (𝑑 = 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝑐))
1312iuneq2d 4579 . . . . 5 (𝑑 = 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
1411, 13eqeq12d 2666 . . . 4 (𝑑 = 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)))
15 suceq 5828 . . . . . 6 (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
1615fveq2d 6233 . . . . 5 (𝑑 = suc 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc suc 𝑐))
17 fveq2 6229 . . . . . 6 (𝑑 = suc 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘suc 𝑐))
1817iuneq2d 4579 . . . . 5 (𝑑 = suc 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
1916, 18eqeq12d 2666 . . . 4 (𝑑 = suc 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
20 suceq 5828 . . . . . 6 (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
2120fveq2d 6233 . . . . 5 (𝑑 = 𝐵 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝐵))
22 fveq2 6229 . . . . . 6 (𝑑 = 𝐵 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝐵))
2322iuneq2d 4579 . . . . 5 (𝑑 = 𝐵 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
2421, 23eqeq12d 2666 . . . 4 (𝑑 = 𝐵 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)))
25 uniiun 4605 . . . . 5 𝑏 = 𝑎𝑏 𝑎
26 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
2726itunisuc 9279 . . . . . 6 ((𝑈𝑏)‘suc ∅) = ((𝑈𝑏)‘∅)
28 vex 3234 . . . . . . . 8 𝑏 ∈ V
2926ituni0 9278 . . . . . . . 8 (𝑏 ∈ V → ((𝑈𝑏)‘∅) = 𝑏)
3028, 29ax-mp 5 . . . . . . 7 ((𝑈𝑏)‘∅) = 𝑏
3130unieqi 4477 . . . . . 6 ((𝑈𝑏)‘∅) = 𝑏
3227, 31eqtri 2673 . . . . 5 ((𝑈𝑏)‘suc ∅) = 𝑏
3326ituni0 9278 . . . . . 6 (𝑎𝑏 → ((𝑈𝑎)‘∅) = 𝑎)
3433iuneq2i 4571 . . . . 5 𝑎𝑏 ((𝑈𝑎)‘∅) = 𝑎𝑏 𝑎
3525, 32, 343eqtr4i 2683 . . . 4 ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)
3626itunisuc 9279 . . . . . 6 ((𝑈𝑏)‘suc suc 𝑐) = ((𝑈𝑏)‘suc 𝑐)
37 unieq 4476 . . . . . . 7 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
3826itunisuc 9279 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
3938a1i 11 . . . . . . . . 9 (𝑎𝑏 → ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐))
4039iuneq2i 4571 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
41 iuncom4 4560 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
4240, 41eqtr2i 2674 . . . . . . 7 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)
4337, 42syl6eq 2701 . . . . . 6 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4436, 43syl5eq 2697 . . . . 5 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4544a1i 11 . . . 4 (𝑐 ∈ ω → (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
469, 14, 19, 24, 35, 45finds 7134 . . 3 (𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
47 iun0 4608 . . . . 5 𝑎𝑏 ∅ = ∅
4847eqcomi 2660 . . . 4 ∅ = 𝑎𝑏
49 peano2b 7123 . . . . . 6 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
5026itunifn 9277 . . . . . . . 8 (𝑏 ∈ V → (𝑈𝑏) Fn ω)
51 fndm 6028 . . . . . . . 8 ((𝑈𝑏) Fn ω → dom (𝑈𝑏) = ω)
5228, 50, 51mp2b 10 . . . . . . 7 dom (𝑈𝑏) = ω
5352eleq2i 2722 . . . . . 6 (suc 𝐵 ∈ dom (𝑈𝑏) ↔ suc 𝐵 ∈ ω)
5449, 53bitr4i 267 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈𝑏))
55 ndmfv 6256 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝑏) → ((𝑈𝑏)‘suc 𝐵) = ∅)
5654, 55sylnbi 319 . . . 4 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = ∅)
57 vex 3234 . . . . . . . 8 𝑎 ∈ V
5826itunifn 9277 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
59 fndm 6028 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
6057, 58, 59mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
6160eleq2i 2722 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
62 ndmfv 6256 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
6361, 62sylnbir 320 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
6463iuneq2d 4579 . . . 4 𝐵 ∈ ω → 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝑏 ∅)
6548, 56, 643eqtr4a 2711 . . 3 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
6646, 65pm2.61i 176 . 2 ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)
674, 66vtoclg 3297 1 (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948   cuni 4468   ciun 4552  cmpt 4762  dom cdm 5143  cres 5145  suc csuc 5763   Fn wfn 5921  cfv 5926  ωcom 7107  reccrdg 7550
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  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  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-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  hsmexlem4  9289
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