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Theorem itunisuc 9443
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 7685 . . . . . 6 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2 fvex 6342 . . . . . . 7 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3 unieq 4582 . . . . . . . 8 (𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) → 𝑎 = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
4 unieq 4582 . . . . . . . . 9 (𝑦 = 𝑎 𝑦 = 𝑎)
54cbvmptv 4884 . . . . . . . 8 (𝑦 ∈ V ↦ 𝑦) = (𝑎 ∈ V ↦ 𝑎)
62uniex 7100 . . . . . . . 8 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
73, 5, 6fvmpt 6424 . . . . . . 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, 8syl6eq 2821 . . . . 5 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
109adantl 467 . . . 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 9440 . . . . . 6 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
1312fveq1d 6334 . . . . 5 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1413adantr 466 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1512fveq1d 6334 . . . . . 6 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1615adantr 466 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1716unieqd 4584 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1810, 14, 173eqtr4d 2815 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
19 uni0 4601 . . . . 5 ∅ = ∅
2019eqcomi 2780 . . . 4 ∅ =
2111itunifn 9441 . . . . . . . . . 10 (𝐴 ∈ V → (𝑈𝐴) Fn ω)
22 fndm 6130 . . . . . . . . . 10 ((𝑈𝐴) Fn ω → dom (𝑈𝐴) = ω)
2321, 22syl 17 . . . . . . . . 9 (𝐴 ∈ V → dom (𝑈𝐴) = ω)
2423eleq2d 2836 . . . . . . . 8 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ suc 𝐵 ∈ ω))
25 peano2b 7228 . . . . . . . 8 (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
2624, 25syl6bbr 278 . . . . . . 7 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
2726notbid 307 . . . . . 6 (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
2827biimpar 463 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈𝐴))
29 ndmfv 6359 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3028, 29syl 17 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3123eleq2d 2836 . . . . . . . 8 (𝐴 ∈ V → (𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
3231notbid 307 . . . . . . 7 (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
3332biimpar 463 . . . . . 6 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈𝐴))
34 ndmfv 6359 . . . . . 6 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘𝐵) = ∅)
3533, 34syl 17 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3635unieqd 4584 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘𝐵) = ∅)
3720, 30, 363eqtr4a 2831 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
3818, 37pm2.61dan 813 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
39 0fv 6368 . . . . 5 (∅‘𝐵) = ∅
4039unieqi 4583 . . . 4 (∅‘𝐵) =
41 0fv 6368 . . . 4 (∅‘suc 𝐵) = ∅
4219, 40, 413eqtr4ri 2804 . . 3 (∅‘suc 𝐵) = (∅‘𝐵)
43 fvprc 6326 . . . 4 𝐴 ∈ V → (𝑈𝐴) = ∅)
4443fveq1d 6334 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
4543fveq1d 6334 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4645unieqd 4584 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4742, 44, 463eqtr4a 2831 . 2 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
4838, 47pm2.61i 176 1 ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063   cuni 4574  cmpt 4863  dom cdm 5249  cres 5251  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212  reccrdg 7658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659
This theorem is referenced by:  itunitc1  9444  itunitc  9445  ituniiun  9446
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