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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 8046 | . . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | 1 | tfr1a 8030 | . 2 ⊢ (Fun rec(𝐹, 𝐴) ∧ Lim dom rec(𝐹, 𝐴)) |
3 | 2 | simpri 488 | 1 ⊢ Lim dom rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3494 ∅c0 4291 ifcif 4467 ∪ cuni 4838 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 Lim wlim 6192 Fun wfun 6349 ‘cfv 6355 reccrdg 8045 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-wrecs 7947 df-recs 8008 df-rdg 8046 |
This theorem is referenced by: rdg0 8057 rdgsucg 8059 rdglimg 8061 rdgsucmptnf 8065 frfnom 8070 frsuc 8072 r1funlim 9195 ackbij2 9665 |
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