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Theorem ituniiun 9836
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 6663 . . . 4 (𝑏 = 𝐴 → (𝑈𝑏) = (𝑈𝐴))
21fveq1d 6665 . . 3 (𝑏 = 𝐴 → ((𝑈𝑏)‘suc 𝐵) = ((𝑈𝐴)‘suc 𝐵))
3 iuneq1 4926 . . 3 (𝑏 = 𝐴 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
42, 3eqeq12d 2835 . 2 (𝑏 = 𝐴 → (((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵) ↔ ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵)))
5 suceq 6249 . . . . . 6 (𝑑 = ∅ → suc 𝑑 = suc ∅)
65fveq2d 6667 . . . . 5 (𝑑 = ∅ → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc ∅))
7 fveq2 6663 . . . . . 6 (𝑑 = ∅ → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘∅))
87iuneq2d 4939 . . . . 5 (𝑑 = ∅ → 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘∅))
96, 8eqeq12d 2835 . . . 4 (𝑑 = ∅ → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)))
10 suceq 6249 . . . . . 6 (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
1110fveq2d 6667 . . . . 5 (𝑑 = 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝑐))
12 fveq2 6663 . . . . . 6 (𝑑 = 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝑐))
1312iuneq2d 4939 . . . . 5 (𝑑 = 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
1411, 13eqeq12d 2835 . . . 4 (𝑑 = 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)))
15 suceq 6249 . . . . . 6 (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
1615fveq2d 6667 . . . . 5 (𝑑 = suc 𝑐 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc suc 𝑐))
17 fveq2 6663 . . . . . 6 (𝑑 = suc 𝑐 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘suc 𝑐))
1817iuneq2d 4939 . . . . 5 (𝑑 = suc 𝑐 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
1916, 18eqeq12d 2835 . . . 4 (𝑑 = suc 𝑐 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)))
20 suceq 6249 . . . . . 6 (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
2120fveq2d 6667 . . . . 5 (𝑑 = 𝐵 → ((𝑈𝑏)‘suc 𝑑) = ((𝑈𝑏)‘suc 𝐵))
22 fveq2 6663 . . . . . 6 (𝑑 = 𝐵 → ((𝑈𝑎)‘𝑑) = ((𝑈𝑎)‘𝐵))
2322iuneq2d 4939 . . . . 5 (𝑑 = 𝐵 𝑎𝑏 ((𝑈𝑎)‘𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
2421, 23eqeq12d 2835 . . . 4 (𝑑 = 𝐵 → (((𝑈𝑏)‘suc 𝑑) = 𝑎𝑏 ((𝑈𝑎)‘𝑑) ↔ ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)))
25 uniiun 4973 . . . . 5 𝑏 = 𝑎𝑏 𝑎
26 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
2726itunisuc 9833 . . . . . 6 ((𝑈𝑏)‘suc ∅) = ((𝑈𝑏)‘∅)
2826ituni0 9832 . . . . . . . 8 (𝑏 ∈ V → ((𝑈𝑏)‘∅) = 𝑏)
2928elv 3498 . . . . . . 7 ((𝑈𝑏)‘∅) = 𝑏
3029unieqi 4839 . . . . . 6 ((𝑈𝑏)‘∅) = 𝑏
3127, 30eqtri 2842 . . . . 5 ((𝑈𝑏)‘suc ∅) = 𝑏
3226ituni0 9832 . . . . . 6 (𝑎𝑏 → ((𝑈𝑎)‘∅) = 𝑎)
3332iuneq2i 4931 . . . . 5 𝑎𝑏 ((𝑈𝑎)‘∅) = 𝑎𝑏 𝑎
3425, 31, 333eqtr4i 2852 . . . 4 ((𝑈𝑏)‘suc ∅) = 𝑎𝑏 ((𝑈𝑎)‘∅)
3526itunisuc 9833 . . . . . 6 ((𝑈𝑏)‘suc suc 𝑐) = ((𝑈𝑏)‘suc 𝑐)
36 unieq 4838 . . . . . . 7 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐))
3726itunisuc 9833 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
3837a1i 11 . . . . . . . . 9 (𝑎𝑏 → ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐))
3938iuneq2i 4931 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
40 iuncom4 4918 . . . . . . . 8 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐)
4139, 40eqtr2i 2843 . . . . . . 7 𝑎𝑏 ((𝑈𝑎)‘𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐)
4236, 41syl6eq 2870 . . . . . 6 (((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘𝑐) → ((𝑈𝑏)‘suc 𝑐) = 𝑎𝑏 ((𝑈𝑎)‘suc 𝑐))
4335, 42syl5eq 2866 . . . . 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 7600 . . 3 (𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
46 iun0 4976 . . . . 5 𝑎𝑏 ∅ = ∅
4746eqcomi 2828 . . . 4 ∅ = 𝑎𝑏
48 peano2b 7588 . . . . . 6 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
49 vex 3496 . . . . . . . 8 𝑏 ∈ V
5026itunifn 9831 . . . . . . . 8 (𝑏 ∈ V → (𝑈𝑏) Fn ω)
51 fndm 6448 . . . . . . . 8 ((𝑈𝑏) Fn ω → dom (𝑈𝑏) = ω)
5249, 50, 51mp2b 10 . . . . . . 7 dom (𝑈𝑏) = ω
5352eleq2i 2902 . . . . . 6 (suc 𝐵 ∈ dom (𝑈𝑏) ↔ suc 𝐵 ∈ ω)
5448, 53bitr4i 280 . . . . 5 (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈𝑏))
55 ndmfv 6693 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝑏) → ((𝑈𝑏)‘suc 𝐵) = ∅)
5654, 55sylnbi 332 . . . 4 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = ∅)
57 vex 3496 . . . . . . . 8 𝑎 ∈ V
5826itunifn 9831 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
59 fndm 6448 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
6057, 58, 59mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
6160eleq2i 2902 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
62 ndmfv 6693 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
6361, 62sylnbir 333 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
6463iuneq2d 4939 . . . 4 𝐵 ∈ ω → 𝑎𝑏 ((𝑈𝑎)‘𝐵) = 𝑎𝑏 ∅)
6547, 56, 643eqtr4a 2880 . . 3 𝐵 ∈ ω → ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵))
6645, 65pm2.61i 183 . 2 ((𝑈𝑏)‘suc 𝐵) = 𝑎𝑏 ((𝑈𝑎)‘𝐵)
674, 66vtoclg 3566 1 (𝐴𝑉 → ((𝑈𝐴)‘suc 𝐵) = 𝑎𝐴 ((𝑈𝑎)‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1530  wcel 2107  Vcvv 3493  c0 4289   cuni 4830   ciun 4910  cmpt 5137  dom cdm 5548  cres 5550  suc csuc 6186   Fn wfn 6343  cfv 6348  ωcom 7572  reccrdg 8037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038
This theorem is referenced by:  hsmexlem4  9843
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