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Theorem itunisuc 10414
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 8437 . . . . . 6 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2 fvex 6905 . . . . . . 7 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3 unieq 4920 . . . . . . . 8 (𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) → 𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
4 unieq 4920 . . . . . . . . 9 (𝑦 = 𝑎 𝑦 = 𝑎)
54cbvmptv 5262 . . . . . . . 8 (𝑦 ∈ V ↦ 𝑦) = (𝑎 ∈ V ↦ 𝑎)
62uniex 7731 . . . . . . . 8 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
73, 5, 6fvmpt 6999 . . . . . . 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 2789 . . . . 5 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
109adantl 483 . . . 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 10411 . . . . . 6 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
1312fveq1d 6894 . . . . 5 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1413adantr 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1512fveq1d 6894 . . . . . 6 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1615adantr 482 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1716unieqd 4923 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1810, 14, 173eqtr4d 2783 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
19 uni0 4940 . . . . 5 ∅ = ∅
2019eqcomi 2742 . . . 4 ∅ =
2111itunifn 10412 . . . . . . . . . 10 (𝐴 ∈ V → (𝑈𝐴) Fn ω)
2221fndmd 6655 . . . . . . . . 9 (𝐴 ∈ V → dom (𝑈𝐴) = ω)
2322eleq2d 2820 . . . . . . . 8 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ suc 𝐵 ∈ ω))
24 peano2b 7872 . . . . . . . 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 479 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈𝐴))
28 ndmfv 6927 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘suc 𝐵) = ∅)
2927, 28syl 17 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3022eleq2d 2820 . . . . . . . 8 (𝐴 ∈ V → (𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
3130notbid 318 . . . . . . 7 (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
3231biimpar 479 . . . . . 6 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈𝐴))
33 ndmfv 6927 . . . . . 6 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘𝐵) = ∅)
3432, 33syl 17 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3534unieqd 4923 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3620, 29, 353eqtr4a 2799 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
3718, 36pm2.61dan 812 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
38 0fv 6936 . . . . 5 (∅‘𝐵) = ∅
3938unieqi 4922 . . . 4 (∅‘𝐵) =
40 0fv 6936 . . . 4 (∅‘suc 𝐵) = ∅
4119, 39, 403eqtr4ri 2772 . . 3 (∅‘suc 𝐵) = (∅‘𝐵)
42 fvprc 6884 . . . 4 𝐴 ∈ V → (𝑈𝐴) = ∅)
4342fveq1d 6894 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
4442fveq1d 6894 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4544unieqd 4923 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4641, 43, 453eqtr4a 2799 . 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 397   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323   cuni 4909  cmpt 5232  dom cdm 5677  cres 5679  suc csuc 6367  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:  itunitc1  10415  itunitc  10416  ituniiun  10417
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