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Theorem r1lim 9530

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 6329	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
2		r1fnon 9525	. . . 4 ⊢ 𝑅₁ Fn On
3		fndm 6536	. . . 4 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
4	2, 3	ax-mp 5	. . 3 ⊢ dom 𝑅₁ = On
5	1, 4	eleqtrrdi 2850	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
6		r1limg 9529	. 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 396 = wceq 1539 ∈ wcel 2106 ∪ ciun 4924 dom cdm 5589 Oncon0 6266 Lim wlim 6267 Fn wfn 6428 ‘cfv 6433 𝑅₁cr1 9520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-r1 9522

This theorem is referenced by: r1sdom 9532 r1om 10000 inar1 10531 inatsk 10534 grur1a 10575 grur1cld 41850

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