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Theorem itunisuc 10416
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 8439 . . . . . 6 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2 fvex 6904 . . . . . . 7 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3 unieq 4919 . . . . . . . 8 (𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) → 𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
4 unieq 4919 . . . . . . . . 9 (𝑦 = 𝑎 𝑦 = 𝑎)
54cbvmptv 5261 . . . . . . . 8 (𝑦 ∈ V ↦ 𝑦) = (𝑎 ∈ V ↦ 𝑎)
62uniex 7733 . . . . . . . 8 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
73, 5, 6fvmpt 6998 . . . . . . 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 2788 . . . . 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 10413 . . . . . 6 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
1312fveq1d 6893 . . . . 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 6893 . . . . . 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 4922 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1810, 14, 173eqtr4d 2782 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
19 uni0 4939 . . . . 5 ∅ = ∅
2019eqcomi 2741 . . . 4 ∅ =
2111itunifn 10414 . . . . . . . . . 10 (𝐴 ∈ V → (𝑈𝐴) Fn ω)
2221fndmd 6654 . . . . . . . . 9 (𝐴 ∈ V → dom (𝑈𝐴) = ω)
2322eleq2d 2819 . . . . . . . 8 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ suc 𝐵 ∈ ω))
24 peano2b 7874 . . . . . . . 8 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
2523, 24bitr4di 288 . . . . . . 7 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
2625notbid 317 . . . . . 6 (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
2726biimpar 478 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈𝐴))
28 ndmfv 6926 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘suc 𝐵) = ∅)
2927, 28syl 17 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3022eleq2d 2819 . . . . . . . 8 (𝐴 ∈ V → (𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
3130notbid 317 . . . . . . 7 (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
3231biimpar 478 . . . . . 6 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈𝐴))
33 ndmfv 6926 . . . . . 6 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘𝐵) = ∅)
3432, 33syl 17 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3534unieqd 4922 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3620, 29, 353eqtr4a 2798 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
3718, 36pm2.61dan 811 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
38 0fv 6935 . . . . 5 (∅‘𝐵) = ∅
3938unieqi 4921 . . . 4 (∅‘𝐵) =
40 0fv 6935 . . . 4 (∅‘suc 𝐵) = ∅
4119, 39, 403eqtr4ri 2771 . . 3 (∅‘suc 𝐵) = (∅‘𝐵)
42 fvprc 6883 . . . 4 𝐴 ∈ V → (𝑈𝐴) = ∅)
4342fveq1d 6893 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
4442fveq1d 6893 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4544unieqd 4922 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4641, 43, 453eqtr4a 2798 . 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 1541  wcel 2106  Vcvv 3474  c0 4322   cuni 4908  cmpt 5231  dom cdm 5676  cres 5678  suc csuc 6366  cfv 6543  ωcom 7857  reccrdg 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412
This theorem is referenced by:  itunitc1  10417  itunitc  10418  ituniiun  10419
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