MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itunisuc Structured version   Visualization version   GIF version

Theorem itunisuc 10163
Description: Successor iterated union. (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
itunisuc ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunisuc
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 frsuc 8256 . . . . . 6 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2 fvex 6780 . . . . . . 7 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3 unieq 4851 . . . . . . . 8 (𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) → 𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
4 unieq 4851 . . . . . . . . 9 (𝑦 = 𝑎 𝑦 = 𝑎)
54cbvmptv 5187 . . . . . . . 8 (𝑦 ∈ V ↦ 𝑦) = (𝑎 ∈ V ↦ 𝑎)
62uniex 7585 . . . . . . . 8 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
73, 5, 6fvmpt 6868 . . . . . . 7 (((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V → ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
82, 7ax-mp 5 . . . . . 6 ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)
91, 8eqtrdi 2794 . . . . 5 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
109adantl 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
11 ituni.u . . . . . . 7 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
1211itunifval 10160 . . . . . 6 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
1312fveq1d 6769 . . . . 5 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1413adantr 481 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1512fveq1d 6769 . . . . . 6 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1615adantr 481 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1716unieqd 4854 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1810, 14, 173eqtr4d 2788 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
19 uni0 4870 . . . . 5 ∅ = ∅
2019eqcomi 2747 . . . 4 ∅ =
2111itunifn 10161 . . . . . . . . . 10 (𝐴 ∈ V → (𝑈𝐴) Fn ω)
2221fndmd 6531 . . . . . . . . 9 (𝐴 ∈ V → dom (𝑈𝐴) = ω)
2322eleq2d 2824 . . . . . . . 8 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ suc 𝐵 ∈ ω))
24 peano2b 7720 . . . . . . . 8 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
2523, 24bitr4di 289 . . . . . . 7 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
2625notbid 318 . . . . . 6 (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
2726biimpar 478 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈𝐴))
28 ndmfv 6797 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘suc 𝐵) = ∅)
2927, 28syl 17 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3022eleq2d 2824 . . . . . . . 8 (𝐴 ∈ V → (𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
3130notbid 318 . . . . . . 7 (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
3231biimpar 478 . . . . . 6 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈𝐴))
33 ndmfv 6797 . . . . . 6 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘𝐵) = ∅)
3432, 33syl 17 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3534unieqd 4854 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3620, 29, 353eqtr4a 2804 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
3718, 36pm2.61dan 810 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
38 0fv 6806 . . . . 5 (∅‘𝐵) = ∅
3938unieqi 4853 . . . 4 (∅‘𝐵) =
40 0fv 6806 . . . 4 (∅‘suc 𝐵) = ∅
4119, 39, 403eqtr4ri 2777 . . 3 (∅‘suc 𝐵) = (∅‘𝐵)
42 fvprc 6759 . . . 4 𝐴 ∈ V → (𝑈𝐴) = ∅)
4342fveq1d 6769 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
4442fveq1d 6769 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4544unieqd 4854 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4641, 43, 453eqtr4a 2804 . 2 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
4737, 46pm2.61i 182 1 ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wcel 2106  Vcvv 3430  c0 4257   cuni 4840  cmpt 5157  dom cdm 5585  cres 5587  suc csuc 6262  cfv 6427  ωcom 7703  reccrdg 8228
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 5222  ax-nul 5229  ax-pr 5351  ax-un 7579  ax-inf2 9387
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 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-om 7704  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229
This theorem is referenced by:  itunitc1  10164  itunitc  10165  ituniiun  10166
  Copyright terms: Public domain W3C validator