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Mirrors > Home > MPE Home > Th. List > rdgdmlim | Structured version Visualization version GIF version |
Description: The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdgdmlim | ⊢ Lim dom rec(𝐹, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rdg 8425 | . . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | 1 | tfr1a 8409 | . 2 ⊢ (Fun rec(𝐹, 𝐴) ∧ Lim dom rec(𝐹, 𝐴)) |
3 | 2 | simpri 485 | 1 ⊢ Lim dom rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 Vcvv 3470 ∅c0 4319 ifcif 4525 ∪ cuni 4904 ↦ cmpt 5226 dom cdm 5673 ran crn 5674 Lim wlim 6365 Fun wfun 6537 ‘cfv 6543 reccrdg 8424 |
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 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 |
This theorem is referenced by: rdg0 8436 rdgsucg 8438 rdglimg 8440 rdgsucmptnf 8444 frfnom 8450 frsuc 8452 r1funlim 9784 ackbij2 10261 bj-rdg0gALT 36545 |
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