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Theorem itunisuc 10444
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 8458 . . . . . 6 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ 𝑦)‘((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2 fvex 6909 . . . . . . 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 7747 . . . . . . . 8 ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
73, 5, 6fvmpt 7004 . . . . . . 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 2781 . . . . 5 (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
109adantl 480 . . . 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 10441 . . . . . 6 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
1312fveq1d 6898 . . . . 5 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1413adantr 479 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
1512fveq1d 6898 . . . . . 6 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω)‘𝐵))
1615adantr 479 . . . . 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 2775 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
19 uni0 4939 . . . . 5 ∅ = ∅
2019eqcomi 2734 . . . 4 ∅ =
2111itunifn 10442 . . . . . . . . . 10 (𝐴 ∈ V → (𝑈𝐴) Fn ω)
2221fndmd 6660 . . . . . . . . 9 (𝐴 ∈ V → dom (𝑈𝐴) = ω)
2322eleq2d 2811 . . . . . . . 8 (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈𝐴) ↔ suc 𝐵 ∈ ω))
24 peano2b 7888 . . . . . . . 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 476 . . . . 5 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈𝐴))
28 ndmfv 6931 . . . . 5 (¬ suc 𝐵 ∈ dom (𝑈𝐴) → ((𝑈𝐴)‘suc 𝐵) = ∅)
2927, 28syl 17 . . . 4 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ∅)
3022eleq2d 2811 . . . . . . . 8 (𝐴 ∈ V → (𝐵 ∈ dom (𝑈𝐴) ↔ 𝐵 ∈ ω))
3130notbid 317 . . . . . . 7 (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈𝐴) ↔ ¬ 𝐵 ∈ ω))
3231biimpar 476 . . . . . 6 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈𝐴))
33 ndmfv 6931 . . . . . 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 2791 . . 3 ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
3718, 36pm2.61dan 811 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = ((𝑈𝐴)‘𝐵))
38 0fv 6940 . . . . 5 (∅‘𝐵) = ∅
3938unieqi 4921 . . . 4 (∅‘𝐵) =
40 0fv 6940 . . . 4 (∅‘suc 𝐵) = ∅
4119, 39, 403eqtr4ri 2764 . . 3 (∅‘suc 𝐵) = (∅‘𝐵)
42 fvprc 6888 . . . 4 𝐴 ∈ V → (𝑈𝐴) = ∅)
4342fveq1d 6898 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
4442fveq1d 6898 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4544unieqd 4922 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
4641, 43, 453eqtr4a 2791 . 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 394   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322   cuni 4909  cmpt 5232  dom cdm 5678  cres 5680  suc csuc 6373  cfv 6549  ωcom 7871  reccrdg 8430
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431
This theorem is referenced by:  itunitc1  10445  itunitc  10446  ituniiun  10447
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