Theorem r1limg 8517

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 8510	. . . . 5 ⊢ 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
2	1	dmeqi 5247	. . . 4 ⊢ dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
3	2	eleq2i 2680	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4		rdglimg 7408	. . 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 488	. 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴))
6	1	fveq1i 6104	. 2 ⊢ (𝑅1‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7		r1funlim 8512	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
8	7	simpli 473	. . . 4 ⊢ Fun 𝑅1
9		funiunfv 6410	. . . 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 5382	. . . 4 ⊢ (𝑅1 “ 𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
12	11	unieqi 4381	. . 3 ⊢ ∪ (𝑅1 “ 𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
13	10, 12	eqtri 2632	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
14	5, 6, 13	3eqtr4g 2669	1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  dom cdm 5038   “ cima 5041  Lim wlim 5641  Fun wfun 5798  ‘cfv 5804  reccrdg 7392  𝑅₁cr1 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510

This theorem is referenced by:  r1lim  8518  r1tr  8522  r1ordg  8524  r1pwss  8530  r1val1  8532