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Theorem ituniiun 10417
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 6892 . . . 4 (𝑏 = 𝐴 → (𝑈𝑏) = (𝑈𝐴))
21fveq1d 6894 . . 3 (𝑏 = 𝐴 → ((𝑈𝑏)‘suc 𝐵) = ((𝑈𝐴)‘suc 𝐵))
3 iuneq1 5014 . . 3 (𝑏 = 𝐴 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
42, 3eqeq12d 2749 . 2 (𝑏 = 𝐴 → (((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵) ↔ ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵)))
5 suceq 6431 . . . . . 6 (𝑑 = ∅ → suc 𝑑 = suc ∅)
65fveq2d 6896 . . . . 5 (𝑑 = ∅ → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc ∅))
7 fveq2 6892 . . . . . 6 (𝑑 = ∅ → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘∅))
87iuneq2d 5027 . . . . 5 (𝑑 = ∅ → 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘∅))
96, 8eqeq12d 2749 . . . 4 (𝑑 = ∅ → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)))
10 suceq 6431 . . . . . 6 (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
1110fveq2d 6896 . . . . 5 (𝑑 = 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝑐))
12 fveq2 6892 . . . . . 6 (𝑑 = 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝑐))
1312iuneq2d 5027 . . . . 5 (𝑑 = 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
1411, 13eqeq12d 2749 . . . 4 (𝑑 = 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)))
15 suceq 6431 . . . . . 6 (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
1615fveq2d 6896 . . . . 5 (𝑑 = suc 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc suc 𝑐))
17 fveq2 6892 . . . . . 6 (𝑑 = suc 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘suc 𝑐))
1817iuneq2d 5027 . . . . 5 (𝑑 = suc 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
1916, 18eqeq12d 2749 . . . 4 (𝑑 = suc 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
20 suceq 6431 . . . . . 6 (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
2120fveq2d 6896 . . . . 5 (𝑑 = 𝐵 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝐵))
22 fveq2 6892 . . . . . 6 (𝑑 = 𝐵 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝐵))
2322iuneq2d 5027 . . . . 5 (𝑑 = 𝐵 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
2421, 23eqeq12d 2749 . . . 4 (𝑑 = 𝐵 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)))
25 uniiun 5062 . . . . 5 𝑏 = 𝑎𝑏 𝑎
26 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
2726itunisuc 10414 . . . . . 6 ((𝑈𝑏)‘suc ∅) = ((𝑈𝑏)‘∅)
2826ituni0 10413 . . . . . . . 8 (𝑏 ∈ V → ((𝑈𝑏)‘∅) = 𝑏)
2928elv 3481 . . . . . . 7 ((𝑈𝑏)‘∅) = 𝑏
3029unieqi 4922 . . . . . 6 ((𝑈𝑏)‘∅) = 𝑏
3127, 30eqtri 2761 . . . . 5 ((𝑈𝑏)‘suc ∅) = 𝑏
3226ituni0 10413 . . . . . 6 (𝑎𝑏 → ((𝑈𝑎)‘∅) = 𝑎)
3332iuneq2i 5019 . . . . 5 𝑎𝑏 ((𝑈𝑎)‘∅) = 𝑎𝑏 𝑎
3425, 31, 333eqtr4i 2771 . . . 4 ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)
3526itunisuc 10414 . . . . . 6 ((𝑈𝑏)‘suc suc 𝑐) = ((𝑈𝑏)‘suc 𝑐)
36 unieq 4920 . . . . . . 7 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
3726itunisuc 10414 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
3837a1i 11 . . . . . . . . 9 (𝑎𝑏 → ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐))
3938iuneq2i 5019 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
40 iuncom4 5006 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
4139, 40eqtr2i 2762 . . . . . . 7 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)
4236, 41eqtrdi 2789 . . . . . 6 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4335, 42eqtrid 2785 . . . . 5 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4443a1i 11 . . . 4 (𝑐 ∈ ω → (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
459, 14, 19, 24, 34, 44finds 7889 . . 3 (𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
46 iun0 5066 . . . . 5 𝑎𝑏 ∅ = ∅
4746eqcomi 2742 . . . 4 ∅ = 𝑎𝑏
48 peano2b 7872 . . . . . 6 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
49 vex 3479 . . . . . . . 8 𝑏 ∈ V
5026itunifn 10412 . . . . . . . 8 (𝑏 ∈ V → (𝑈𝑏) Fn ω)
51 fndm 6653 . . . . . . . 8 ((𝑈𝑏) Fn ω → dom (𝑈𝑏) = ω)
5249, 50, 51mp2b 10 . . . . . . 7 dom (𝑈𝑏) = ω
5352eleq2i 2826 . . . . . 6 (suc 𝐵 ∈ dom (𝑈𝑏) ↔ suc 𝐵 ∈ ω)
5448, 53bitr4i 278 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈𝑏))
55 ndmfv 6927 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝑏) → ((𝑈𝑏)‘suc 𝐵) = ∅)
5654, 55sylnbi 330 . . . 4 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = ∅)
57 vex 3479 . . . . . . . 8 𝑎 ∈ V
5826itunifn 10412 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
59 fndm 6653 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
6057, 58, 59mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
6160eleq2i 2826 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
62 ndmfv 6927 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
6361, 62sylnbir 331 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
6463iuneq2d 5027 . . . 4 𝐵 ∈ ω → 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝑏 ∅)
6547, 56, 643eqtr4a 2799 . . 3 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
6645, 65pm2.61i 182 . 2 ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)
674, 66vtoclg 3557 1 (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323   cuni 4909   ciun 4998  cmpt 5232  dom cdm 5677  cres 5679  suc csuc 6367   Fn wfn 6539  cfv 6544  ωcom 7855  reccrdg 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410
This theorem is referenced by:  hsmexlem4  10424
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