Theorem r1limg 9686

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 9679	. . . . 5 ⊢ 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
2	1	dmeqi 5851	. . . 4 ⊢ dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
3	2	eleq2i 2820	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4		rdglimg 8354	. . 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 581	. 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
6	1	fveq1i 6827	. 2 ⊢ (𝑅1‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7		r1funlim 9681	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
8	7	simpli 483	. . . 4 ⊢ Fun 𝑅1
9		funiunfv 7188	. . . 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 6012	. . . 4 ⊢ (𝑅1 “ 𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
12	11	unieqi 4873	. . 3 ⊢ ∪ (𝑅1 “ 𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
13	10, 12	eqtri 2752	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
14	5, 6, 13	3eqtr4g 2789	1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  𝒫 cpw 4553  ∪ cuni 4861  ∪ ciun 4944   ↦ cmpt 5176  dom cdm 5623   “ cima 5626  Lim wlim 6312  Fun wfun 6480  ‘cfv 6486  reccrdg 8338  𝑅₁cr1 9677

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679

This theorem is referenced by:  r1lim  9687  r1tr  9691  r1ordg  9693  r1pwss  9699  r1val1  9701