Theorem r1limg 9768

Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
r1limg	⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem r1limg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-r1 9761	. . . . 5 ⊢ 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
2	1	dmeqi 5903	. . . 4 ⊢ dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
3	2	eleq2i 2823	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4		rdglimg 8427	. . 3 ⊢ ((𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
5	3, 4	sylanb 579	. 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
6	1	fveq1i 6891	. 2 ⊢ (𝑅1‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7		r1funlim 9763	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
8	7	simpli 482	. . . 4 ⊢ Fun 𝑅1
9		funiunfv 7249	. . . 4 ⊢ (Fun 𝑅1 → ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (𝑅1 “ 𝐴))
10	8, 9	ax-mp 5	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (𝑅1 “ 𝐴)
11	1	imaeq1i 6055	. . . 4 ⊢ (𝑅1 “ 𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
12	11	unieqi 4920	. . 3 ⊢ ∪ (𝑅1 “ 𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
13	10, 12	eqtri 2758	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
14	5, 6, 13	3eqtr4g 2795	1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ∅c0 4321  𝒫 cpw 4601  ∪ cuni 4907  ∪ ciun 4996   ↦ cmpt 5230  dom cdm 5675   “ cima 5678  Lim wlim 6364  Fun wfun 6536  ‘cfv 6542  reccrdg 8411  𝑅₁cr1 9759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761

This theorem is referenced by:  r1lim  9769  r1tr  9773  r1ordg  9775  r1pwss  9781  r1val1  9783