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Theorem itunifval 9573
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 3414 . 2 (𝐴𝑉𝐴 ∈ V)
2 rdgeq2 7791 . . . 4 (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ 𝑦), 𝐴))
32reseq1d 5641 . . 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 7795 . . . 4 Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴)
6 omex 8837 . . . 4 ω ∈ V
7 resfunexg 6751 . . . 4 ((Fun rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V)
85, 6, 7mp2an 682 . . 3 (rec((𝑦 ∈ V ↦ 𝑦), 𝐴) ↾ ω) ∈ V
93, 4, 8fvmpt 6542 . 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 1601  wcel 2107  Vcvv 3398   cuni 4671  cmpt 4965  cres 5357  Fun wfun 6129  cfv 6135  ωcom 7343  reccrdg 7788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789
This theorem is referenced by:  itunifn  9574  ituni0  9575  itunisuc  9576
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