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Theorem ituniiun 9497
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 6375 . . . 4 (𝑏 = 𝐴 → (𝑈𝑏) = (𝑈𝐴))
21fveq1d 6377 . . 3 (𝑏 = 𝐴 → ((𝑈𝑏)‘suc 𝐵) = ((𝑈𝐴)‘suc 𝐵))
3 iuneq1 4690 . . 3 (𝑏 = 𝐴 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
42, 3eqeq12d 2780 . 2 (𝑏 = 𝐴 → (((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵) ↔ ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵)))
5 suceq 5973 . . . . . 6 (𝑑 = ∅ → suc 𝑑 = suc ∅)
65fveq2d 6379 . . . . 5 (𝑑 = ∅ → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc ∅))
7 fveq2 6375 . . . . . 6 (𝑑 = ∅ → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘∅))
87iuneq2d 4703 . . . . 5 (𝑑 = ∅ → 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘∅))
96, 8eqeq12d 2780 . . . 4 (𝑑 = ∅ → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)))
10 suceq 5973 . . . . . 6 (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
1110fveq2d 6379 . . . . 5 (𝑑 = 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝑐))
12 fveq2 6375 . . . . . 6 (𝑑 = 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝑐))
1312iuneq2d 4703 . . . . 5 (𝑑 = 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
1411, 13eqeq12d 2780 . . . 4 (𝑑 = 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)))
15 suceq 5973 . . . . . 6 (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
1615fveq2d 6379 . . . . 5 (𝑑 = suc 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc suc 𝑐))
17 fveq2 6375 . . . . . 6 (𝑑 = suc 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘suc 𝑐))
1817iuneq2d 4703 . . . . 5 (𝑑 = suc 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
1916, 18eqeq12d 2780 . . . 4 (𝑑 = suc 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
20 suceq 5973 . . . . . 6 (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
2120fveq2d 6379 . . . . 5 (𝑑 = 𝐵 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝐵))
22 fveq2 6375 . . . . . 6 (𝑑 = 𝐵 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝐵))
2322iuneq2d 4703 . . . . 5 (𝑑 = 𝐵 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
2421, 23eqeq12d 2780 . . . 4 (𝑑 = 𝐵 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)))
25 uniiun 4729 . . . . 5 𝑏 = 𝑎𝑏 𝑎
26 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
2726itunisuc 9494 . . . . . 6 ((𝑈𝑏)‘suc ∅) = ((𝑈𝑏)‘∅)
28 vex 3353 . . . . . . . 8 𝑏 ∈ V
2926ituni0 9493 . . . . . . . 8 (𝑏 ∈ V → ((𝑈𝑏)‘∅) = 𝑏)
3028, 29ax-mp 5 . . . . . . 7 ((𝑈𝑏)‘∅) = 𝑏
3130unieqi 4603 . . . . . 6 ((𝑈𝑏)‘∅) = 𝑏
3227, 31eqtri 2787 . . . . 5 ((𝑈𝑏)‘suc ∅) = 𝑏
3326ituni0 9493 . . . . . 6 (𝑎𝑏 → ((𝑈𝑎)‘∅) = 𝑎)
3433iuneq2i 4695 . . . . 5 𝑎𝑏 ((𝑈𝑎)‘∅) = 𝑎𝑏 𝑎
3525, 32, 343eqtr4i 2797 . . . 4 ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)
3626itunisuc 9494 . . . . . 6 ((𝑈𝑏)‘suc suc 𝑐) = ((𝑈𝑏)‘suc 𝑐)
37 unieq 4602 . . . . . . 7 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
3826itunisuc 9494 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
3938a1i 11 . . . . . . . . 9 (𝑎𝑏 → ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐))
4039iuneq2i 4695 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
41 iuncom4 4684 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
4240, 41eqtr2i 2788 . . . . . . 7 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)
4337, 42syl6eq 2815 . . . . . 6 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4436, 43syl5eq 2811 . . . . 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 7290 . . 3 (𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
47 iun0 4732 . . . . 5 𝑎𝑏 ∅ = ∅
4847eqcomi 2774 . . . 4 ∅ = 𝑎𝑏
49 peano2b 7279 . . . . . 6 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
5026itunifn 9492 . . . . . . . 8 (𝑏 ∈ V → (𝑈𝑏) Fn ω)
51 fndm 6168 . . . . . . . 8 ((𝑈𝑏) Fn ω → dom (𝑈𝑏) = ω)
5228, 50, 51mp2b 10 . . . . . . 7 dom (𝑈𝑏) = ω
5352eleq2i 2836 . . . . . 6 (suc 𝐵 ∈ dom (𝑈𝑏) ↔ suc 𝐵 ∈ ω)
5449, 53bitr4i 269 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈𝑏))
55 ndmfv 6405 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝑏) → ((𝑈𝑏)‘suc 𝐵) = ∅)
5654, 55sylnbi 321 . . . 4 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = ∅)
57 vex 3353 . . . . . . . 8 𝑎 ∈ V
5826itunifn 9492 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
59 fndm 6168 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
6057, 58, 59mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
6160eleq2i 2836 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
62 ndmfv 6405 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
6361, 62sylnbir 322 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
6463iuneq2d 4703 . . . 4 𝐵 ∈ ω → 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝑏 ∅)
6548, 56, 643eqtr4a 2825 . . 3 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
6646, 65pm2.61i 176 . 2 ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)
674, 66vtoclg 3418 1 (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079   cuni 4594   ciun 4676  cmpt 4888  dom cdm 5277  cres 5279  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  reccrdg 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710
This theorem is referenced by:  hsmexlem4  9504
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