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Theorem itunifval 9430
 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 3352 . 2 (𝐴𝑉𝐴 ∈ V)
2 rdgeq2 7677 . . . 4 (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ 𝑦), 𝐴))
32reseq1d 5550 . . 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 7681 . . . 4 Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴)
6 omex 8713 . . . 4 ω ∈ V
7 resfunexg 6643 . . . 4 ((Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V)
85, 6, 7mp2an 710 . . 3 (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V
93, 4, 8fvmpt 6444 . 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 1632   ∈ wcel 2139  Vcvv 3340  ∪ cuni 4588   ↦ cmpt 4881   ↾ cres 5268  Fun wfun 6043  ‘cfv 6049  ωcom 7230  reccrdg 7674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675 This theorem is referenced by:  itunifn  9431  ituni0  9432  itunisuc  9433
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