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Theorem r1lim 9725

Description: Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

**Assertion**
Ref	Expression
r1lim	⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem r1lim**
Step	Hyp	Ref	Expression
1		limelon 6397	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
2		r1fnon 9720	. . . 4 ⊢ 𝑅₁ Fn On
3		fndm 6621	. . . 4 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
4	2, 3	ax-mp 5	. . 3 ⊢ dom 𝑅₁ = On
5	1, 4	eleqtrrdi 2839	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
6		r1limg 9724	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))
7	5, 6	sylancom 588	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∪ ciun 4955 dom cdm 5638 Oncon0 6332 Lim wlim 6333 Fn wfn 6506 ‘cfv 6511 𝑅₁cr1 9715

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-rep 5234 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711

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 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-r1 9717

This theorem is referenced by: r1sdom 9727 r1om 10196 inar1 10728 inatsk 10731 grur1a 10772 grur1cld 44221

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