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Mirrors > Home > MPE Home > Th. List > rdgfnon | Structured version Visualization version GIF version |
Description: The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
rdgfnon | ⊢ rec(𝐹, 𝐴) Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rdg 8311 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | 1 | tfr1 8298 | 1 ⊢ rec(𝐹, 𝐴) Fn On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 Vcvv 3441 ∅c0 4269 ifcif 4473 ∪ cuni 4852 ↦ cmpt 5175 dom cdm 5620 ran crn 5621 Oncon0 6302 Lim wlim 6303 Fn wfn 6474 ‘cfv 6479 reccrdg 8310 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 |
This theorem is referenced by: rdgsuc 8325 rdglim 8327 rdglim2 8333 r1fnon 9624 alephfnon 9922 satfn 33616 rdgprc 34053 |
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