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Theorem itunifval 9831
 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 3462 . 2 (𝐴𝑉𝐴 ∈ V)
2 rdgeq2 8035 . . . 4 (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ 𝑦), 𝐴))
32reseq1d 5821 . . 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 8039 . . . 4 Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴)
6 omex 9094 . . . 4 ω ∈ V
7 resfunexg 6959 . . . 4 ((Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V)
85, 6, 7mp2an 691 . . 3 (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V
93, 4, 8fvmpt 6749 . 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 1538   ∈ wcel 2112  Vcvv 3444  ∪ cuni 4803   ↦ cmpt 5113   ↾ cres 5525  Fun wfun 6322  ‘cfv 6328  ωcom 7564  reccrdg 8032 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033 This theorem is referenced by:  itunifn  9832  ituni0  9833  itunisuc  9834
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