Theorem r1limg 9809

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 9802	. . . . 5 ⊢ 𝑅1 = rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
2	1	dmeqi 5918	. . . 4 ⊢ dom 𝑅1 = dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)
3	2	eleq2i 2831	. . 3 ⊢ (𝐴 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅))
4		rdglimg 8464	. . 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 6908	. 2 ⊢ (𝑅1‘𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅)‘𝐴)
7		r1funlim 9804	. . . . 5 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
8	7	simpli 483	. . . 4 ⊢ Fun 𝑅1
9		funiunfv 7268	. . . 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 6077	. . . 4 ⊢ (𝑅1 “ 𝐴) = (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
12	11	unieqi 4924	. . 3 ⊢ ∪ (𝑅1 “ 𝐴) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
13	10, 12	eqtri 2763	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥) = ∪ (rec((𝑦 ∈ V ↦ 𝒫 𝑦), ∅) “ 𝐴)
14	5, 6, 13	3eqtr4g 2800	1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  𝒫 cpw 4605  ∪ cuni 4912  ∪ ciun 4996   ↦ cmpt 5231  dom cdm 5689   “ cima 5692  Lim wlim 6387  Fun wfun 6557  ‘cfv 6563  reccrdg 8448  𝑅₁cr1 9800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802

This theorem is referenced by:  r1lim  9810  r1tr  9814  r1ordg  9816  r1pwss  9822  r1val1  9824