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Theorem itunifval 10437
Description: Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could conceivably be independently useful. (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
itunifval (𝐴𝑉 → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem itunifval
StepHypRef Expression
1 elex 3482 . 2 (𝐴𝑉𝐴 ∈ V)
2 rdgeq2 8429 . . . 4 (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ 𝑦), 𝐴))
32reseq1d 5976 . . 3 (𝑥 = 𝐴 → (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
4 ituni.u . . 3 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
5 rdgfun 8433 . . . 4 Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴)
6 omex 9664 . . . 4 ω ∈ V
7 resfunexg 7221 . . . 4 ((Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V)
85, 6, 7mp2an 690 . . 3 (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V
93, 4, 8fvmpt 6998 . 2 (𝐴 ∈ V → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
101, 9syl 17 1 (𝐴𝑉 → (𝑈𝐴) = (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3463   cuni 4901  cmpt 5224  cres 5672  Fun wfun 6535  cfv 6541  ωcom 7866  reccrdg 8426
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 5278  ax-sep 5292  ax-nul 5299  ax-pr 5421  ax-un 7736  ax-inf2 9662
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 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7417  df-om 7867  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427
This theorem is referenced by:  itunifn  10438  ituni0  10439  itunisuc  10440
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