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Mirrors > Home > MPE Home > Th. List > frfnom | Structured version Visualization version GIF version |
Description: The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
frfnom | ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 8413 | . . 3 ⊢ Fun rec(𝐹, 𝐴) | |
2 | funres 6588 | . . 3 ⊢ (Fun rec(𝐹, 𝐴) → Fun (rec(𝐹, 𝐴) ↾ ω)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun (rec(𝐹, 𝐴) ↾ ω) |
4 | dmres 6002 | . . 3 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = (ω ∩ dom rec(𝐹, 𝐴)) | |
5 | rdgdmlim 8414 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
6 | limomss 7857 | . . . . 5 ⊢ (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom rec(𝐹, 𝐴) |
8 | df-ss 3965 | . . . 4 ⊢ (ω ⊆ dom rec(𝐹, 𝐴) ↔ (ω ∩ dom rec(𝐹, 𝐴)) = ω) | |
9 | 7, 8 | mpbi 229 | . . 3 ⊢ (ω ∩ dom rec(𝐹, 𝐴)) = ω |
10 | 4, 9 | eqtri 2761 | . 2 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = ω |
11 | df-fn 6544 | . 2 ⊢ ((rec(𝐹, 𝐴) ↾ ω) Fn ω ↔ (Fun (rec(𝐹, 𝐴) ↾ ω) ∧ dom (rec(𝐹, 𝐴) ↾ ω) = ω)) | |
12 | 3, 10, 11 | mpbir2an 710 | 1 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∩ cin 3947 ⊆ wss 3948 dom cdm 5676 ↾ cres 5678 Lim wlim 6363 Fun wfun 6535 Fn wfn 6536 ωcom 7852 reccrdg 8406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 |
This theorem is referenced by: frsucmptn 8436 seqomlem2 8448 seqomlem3 8449 seqomlem4 8450 unblem4 9295 dffi3 9423 inf0 9613 inf3lem6 9625 alephfplem4 10099 alephfp 10100 infpssrlem3 10297 itunifn 10409 hsmexlem5 10422 axdclem2 10512 wunex2 10730 wuncval2 10739 peano5nni 12212 1nn 12220 peano2nn 12221 om2uzrani 13914 om2uzf1oi 13915 uzrdglem 13919 uzrdgfni 13920 uzrdg0i 13921 hashkf 14289 hashgval2 14335 neibastop2lem 35234 |
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