Theorem itunifval 10407

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 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rdgeq2 8408	. . . 4 ⊢ (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴))
3	2	reseq1d 5978	. . 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 8412	. . . 4 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴)
6		omex 9634	. . . 4 ⊢ ω ∈ V
7		resfunexg 7213	. . . 4 ⊢ ((Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V)
8	5, 6, 7	mp2an 690	. . 3 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V
9	3, 4, 8	fvmpt 6995	. 2 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
10	1, 9	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∪ cuni 4907   ↦ cmpt 5230   ↾ cres 5677  Fun wfun 6534  ‘cfv 6540  ωcom 7851  reccrdg 8405

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406

This theorem is referenced by:  itunifn  10408  ituni0  10409  itunisuc  10410