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Theorem r1lim 8808

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 5949	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
2		r1fnon 8803	. . . 4 ⊢ 𝑅₁ Fn On
3		fndm 6151	. . . 4 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
4	2, 3	ax-mp 5	. . 3 ⊢ dom 𝑅₁ = On
5	1, 4	syl6eleqr 2850	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
6		r1limg 8807	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))
7	5, 6	sylancom 704	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∪ ciun 4672 dom cdm 5266 Oncon0 5884 Lim wlim 5885 Fn wfn 6044 ‘cfv 6049 𝑅₁cr1 8798

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-r1 8800

This theorem is referenced by: r1sdom 8810 r1om 9258 inar1 9789 inatsk 9792 grur1a 9833

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