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Theorem r1lim 9684

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 9679	. . . 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 2847	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ dom 𝑅₁)
6		r1limg 9683	. 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 1541 ∈ wcel 2113 ∪ ciun 4946 dom cdm 5624 Oncon0 6317 Lim wlim 6318 Fn wfn 6487 ‘cfv 6492 𝑅₁cr1 9674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7361 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-r1 9676

This theorem is referenced by: r1sdom 9686 r1om 10153 inar1 10686 inatsk 10689 grur1a 10730 r1omfv 35266 grur1cld 44473

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