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Theorem r1lim 9687

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 6382	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
2		r1fnon 9682	. . . 4 ⊢ 𝑅₁ Fn On
3		fndm 6595	. . . 4 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
4	2, 3	ax-mp 5	. . 3 ⊢ dom 𝑅₁ = On
5	1, 4	eleqtrrdi 2848	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
6		r1limg 9686	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))
7	5, 6	sylancom 589	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅₁‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∪ ciun 4934 dom cdm 5624 Oncon0 6317 Lim wlim 6318 Fn wfn 6487 ‘cfv 6492 𝑅₁cr1 9677

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-r1 9679

This theorem is referenced by: r1sdom 9689 r1om 10156 inar1 10689 inatsk 10692 grur1a 10733 r1omfv 35270 grur1cld 44677

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