Theorem itunifval 10370

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 3474	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		rdgeq2 8378	. . . 4 ⊢ (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴))
3	2	reseq1d 5962	. . 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 8382	. . . 4 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴)
6		omex 9595	. . . 4 ⊢ ω ∈ V
7		resfunexg 7195	. . . 4 ⊢ ((Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V)
8	5, 6, 7	mp2an 702	. . 3 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V
9	3, 4, 8	fvmpt 6971	. 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 1559   ∈ wcel 2141  Vcvv 3453  ∪ cuni 4864   ↦ cmpt 5180   ↾ cres 5647  Fun wfun 6511  ‘cfv 6517  ωcom 7842  reccrdg 8375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376

This theorem is referenced by:  itunifn  10371  ituni0  10372  itunisuc  10373