Theorem itunifval 10497

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	⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (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

Step	Hyp	Ref	Expression
1		elex 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rdgeq2 8423	. . . 4 ⊢ (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴))
3	2	reseq1d 5968	. . 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 8427	. . . 4 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴)
6		omex 9647	. . . 4 ⊢ ω ∈ V
7		resfunexg 7221	. . . 4 ⊢ ((Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V)
8	5, 6, 7	mp2an 705	. . 3 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V
9	3, 4, 8	fvmpt 6993	. 2 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
10	1, 9	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∪ cuni 4866   ↦ cmpt 5185   ↾ cres 5652  Fun wfun 6532  ‘cfv 6538  ωcom 7877  reccrdg 8420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421

This theorem is used by:  itunifn  10498  ituni0  10499  itunisuc  10500