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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 8456 | . . 3 ⊢ Fun rec(𝐹, 𝐴) | |
| 2 | funres 6608 | . . 3 ⊢ (Fun rec(𝐹, 𝐴) → Fun (rec(𝐹, 𝐴) ↾ ω)) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun (rec(𝐹, 𝐴) ↾ ω) |
| 4 | dmres 6030 | . . 3 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = (ω ∩ dom rec(𝐹, 𝐴)) | |
| 5 | rdgdmlim 8457 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
| 6 | limomss 7892 | . . . . 5 ⊢ (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴)) | |
| 7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom rec(𝐹, 𝐴) |
| 8 | dfss2 3969 | . . . 4 ⊢ (ω ⊆ dom rec(𝐹, 𝐴) ↔ (ω ∩ dom rec(𝐹, 𝐴)) = ω) | |
| 9 | 7, 8 | mpbi 230 | . . 3 ⊢ (ω ∩ dom rec(𝐹, 𝐴)) = ω |
| 10 | 4, 9 | eqtri 2765 | . 2 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = ω |
| 11 | df-fn 6564 | . 2 ⊢ ((rec(𝐹, 𝐴) ↾ ω) Fn ω ↔ (Fun (rec(𝐹, 𝐴) ↾ ω) ∧ dom (rec(𝐹, 𝐴) ↾ ω) = ω)) | |
| 12 | 3, 10, 11 | mpbir2an 711 | 1 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∩ cin 3950 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 Lim wlim 6385 Fun wfun 6555 Fn wfn 6556 ωcom 7887 reccrdg 8449 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 |
| This theorem is referenced by: frsucmptn 8479 seqomlem2 8491 seqomlem3 8492 seqomlem4 8493 unblem4 9331 dffi3 9471 inf0 9661 inf3lem6 9673 alephfplem4 10147 alephfp 10148 infpssrlem3 10345 itunifn 10457 hsmexlem5 10470 axdclem2 10560 wunex2 10778 wuncval2 10787 peano5nni 12269 1nn 12277 peano2nn 12278 om2uzrani 13993 om2uzf1oi 13994 uzrdglem 13998 uzrdgfni 13999 uzrdg0i 14000 hashkf 14371 hashgval2 14417 noseq0 28296 noseqp1 28297 noseqind 28298 om2noseqfo 28304 noseqrdglem 28311 noseqrdgfn 28312 noseqrdg0 28313 dfnns2 28362 neibastop2lem 36361 |
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