rdgdmlim - Metamath Proof Explorer

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Theorem rdgdmlim 8447

Description: The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)

**Assertion**
Ref	Expression
rdgdmlim	⊢ Lim dom rec(𝐹, 𝐴)

Proof of Theorem rdgdmlim Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rdg 8440	. . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
2	1	tfr1a 8424	. 2 ⊢ (Fun rec(𝐹, 𝐴) ∧ Lim dom rec(𝐹, 𝐴))
3	2	simpri 484	1 ⊢ Lim dom rec(𝐹, 𝐴)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 Vcvv 3462 ∅c0 4325 ifcif 4533 ∪ cuni 4913 ↦ cmpt 5236 dom cdm 5682 ran crn 5683 Lim wlim 6377 Fun wfun 6548 ‘cfv 6554 reccrdg 8439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440

This theorem is referenced by: rdg0 8451 rdgsucg 8453 rdglimg 8455 rdgsucmptnf 8459 frfnom 8465 frsuc 8467 r1funlim 9809 ackbij2 10286 bj-rdg0gALT 36778

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