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Theorem itunifval 10413
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 3487 . 2 (𝐴𝑉𝐴 ∈ V)
2 rdgeq2 8413 . . . 4 (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ 𝑦), 𝐴))
32reseq1d 5974 . . 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 8417 . . . 4 Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴)
6 omex 9640 . . . 4 ω ∈ V
7 resfunexg 7212 . . . 4 ((Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V)
85, 6, 7mp2an 689 . . 3 (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V
93, 4, 8fvmpt 6992 . 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 3468   cuni 4902  cmpt 5224  cres 5671  Fun wfun 6531  cfv 6537  ωcom 7852  reccrdg 8410
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411
This theorem is referenced by:  itunifn  10414  ituni0  10415  itunisuc  10416
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