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Theorem ituniiun 10178
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 6774 . . . 4 (𝑏 = 𝐴 → (𝑈𝑏) = (𝑈𝐴))
21fveq1d 6776 . . 3 (𝑏 = 𝐴 → ((𝑈𝑏)‘suc 𝐵) = ((𝑈𝐴)‘suc 𝐵))
3 iuneq1 4940 . . 3 (𝑏 = 𝐴 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
42, 3eqeq12d 2754 . 2 (𝑏 = 𝐴 → (((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵) ↔ ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵)))
5 suceq 6331 . . . . . 6 (𝑑 = ∅ → suc 𝑑 = suc ∅)
65fveq2d 6778 . . . . 5 (𝑑 = ∅ → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc ∅))
7 fveq2 6774 . . . . . 6 (𝑑 = ∅ → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘∅))
87iuneq2d 4953 . . . . 5 (𝑑 = ∅ → 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘∅))
96, 8eqeq12d 2754 . . . 4 (𝑑 = ∅ → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)))
10 suceq 6331 . . . . . 6 (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
1110fveq2d 6778 . . . . 5 (𝑑 = 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝑐))
12 fveq2 6774 . . . . . 6 (𝑑 = 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝑐))
1312iuneq2d 4953 . . . . 5 (𝑑 = 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
1411, 13eqeq12d 2754 . . . 4 (𝑑 = 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)))
15 suceq 6331 . . . . . 6 (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
1615fveq2d 6778 . . . . 5 (𝑑 = suc 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc suc 𝑐))
17 fveq2 6774 . . . . . 6 (𝑑 = suc 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘suc 𝑐))
1817iuneq2d 4953 . . . . 5 (𝑑 = suc 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
1916, 18eqeq12d 2754 . . . 4 (𝑑 = suc 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
20 suceq 6331 . . . . . 6 (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
2120fveq2d 6778 . . . . 5 (𝑑 = 𝐵 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝐵))
22 fveq2 6774 . . . . . 6 (𝑑 = 𝐵 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝐵))
2322iuneq2d 4953 . . . . 5 (𝑑 = 𝐵 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
2421, 23eqeq12d 2754 . . . 4 (𝑑 = 𝐵 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)))
25 uniiun 4988 . . . . 5 𝑏 = 𝑎𝑏 𝑎
26 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
2726itunisuc 10175 . . . . . 6 ((𝑈𝑏)‘suc ∅) = ((𝑈𝑏)‘∅)
2826ituni0 10174 . . . . . . . 8 (𝑏 ∈ V → ((𝑈𝑏)‘∅) = 𝑏)
2928elv 3438 . . . . . . 7 ((𝑈𝑏)‘∅) = 𝑏
3029unieqi 4852 . . . . . 6 ((𝑈𝑏)‘∅) = 𝑏
3127, 30eqtri 2766 . . . . 5 ((𝑈𝑏)‘suc ∅) = 𝑏
3226ituni0 10174 . . . . . 6 (𝑎𝑏 → ((𝑈𝑎)‘∅) = 𝑎)
3332iuneq2i 4945 . . . . 5 𝑎𝑏 ((𝑈𝑎)‘∅) = 𝑎𝑏 𝑎
3425, 31, 333eqtr4i 2776 . . . 4 ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)
3526itunisuc 10175 . . . . . 6 ((𝑈𝑏)‘suc suc 𝑐) = ((𝑈𝑏)‘suc 𝑐)
36 unieq 4850 . . . . . . 7 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
3726itunisuc 10175 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
3837a1i 11 . . . . . . . . 9 (𝑎𝑏 → ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐))
3938iuneq2i 4945 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
40 iuncom4 4932 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
4139, 40eqtr2i 2767 . . . . . . 7 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)
4236, 41eqtrdi 2794 . . . . . 6 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4335, 42eqtrid 2790 . . . . 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 7745 . . 3 (𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
46 iun0 4991 . . . . 5 𝑎𝑏 ∅ = ∅
4746eqcomi 2747 . . . 4 ∅ = 𝑎𝑏
48 peano2b 7729 . . . . . 6 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
49 vex 3436 . . . . . . . 8 𝑏 ∈ V
5026itunifn 10173 . . . . . . . 8 (𝑏 ∈ V → (𝑈𝑏) Fn ω)
51 fndm 6536 . . . . . . . 8 ((𝑈𝑏) Fn ω → dom (𝑈𝑏) = ω)
5249, 50, 51mp2b 10 . . . . . . 7 dom (𝑈𝑏) = ω
5352eleq2i 2830 . . . . . 6 (suc 𝐵 ∈ dom (𝑈𝑏) ↔ suc 𝐵 ∈ ω)
5448, 53bitr4i 277 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈𝑏))
55 ndmfv 6804 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝑏) → ((𝑈𝑏)‘suc 𝐵) = ∅)
5654, 55sylnbi 330 . . . 4 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = ∅)
57 vex 3436 . . . . . . . 8 𝑎 ∈ V
5826itunifn 10173 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
59 fndm 6536 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
6057, 58, 59mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
6160eleq2i 2830 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
62 ndmfv 6804 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
6361, 62sylnbir 331 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
6463iuneq2d 4953 . . . 4 𝐵 ∈ ω → 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝑏 ∅)
6547, 56, 643eqtr4a 2804 . . 3 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
6645, 65pm2.61i 182 . 2 ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)
674, 66vtoclg 3505 1 (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2106  Vcvv 3432  c0 4256   cuni 4839   ciun 4924  cmpt 5157  dom cdm 5589  cres 5591  suc csuc 6268   Fn wfn 6428  cfv 6433  ωcom 7712  reccrdg 8240
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241
This theorem is referenced by:  hsmexlem4  10185
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