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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 8052 | . . 3 ⊢ Fun rec(𝐹, 𝐴) | |
2 | funres 6397 | . . 3 ⊢ (Fun rec(𝐹, 𝐴) → Fun (rec(𝐹, 𝐴) ↾ ω)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun (rec(𝐹, 𝐴) ↾ ω) |
4 | dmres 5875 | . . 3 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = (ω ∩ dom rec(𝐹, 𝐴)) | |
5 | rdgdmlim 8053 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
6 | limomss 7585 | . . . . 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 3952 | . . . 4 ⊢ (ω ⊆ dom rec(𝐹, 𝐴) ↔ (ω ∩ dom rec(𝐹, 𝐴)) = ω) | |
9 | 7, 8 | mpbi 232 | . . 3 ⊢ (ω ∩ dom rec(𝐹, 𝐴)) = ω |
10 | 4, 9 | eqtri 2844 | . 2 ⊢ dom (rec(𝐹, 𝐴) ↾ ω) = ω |
11 | df-fn 6358 | . 2 ⊢ ((rec(𝐹, 𝐴) ↾ ω) Fn ω ↔ (Fun (rec(𝐹, 𝐴) ↾ ω) ∧ dom (rec(𝐹, 𝐴) ↾ ω) = ω)) | |
12 | 3, 10, 11 | mpbir2an 709 | 1 ⊢ (rec(𝐹, 𝐴) ↾ ω) Fn ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3935 ⊆ wss 3936 dom cdm 5555 ↾ cres 5557 Lim wlim 6192 Fun wfun 6349 Fn wfn 6350 ωcom 7580 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-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 |
This theorem is referenced by: frsucmptn 8074 seqomlem2 8087 seqomlem3 8088 seqomlem4 8089 unblem4 8773 dffi3 8895 inf0 9084 inf3lem6 9096 alephfplem4 9533 alephfp 9534 infpssrlem3 9727 itunifn 9839 hsmexlem5 9852 axdclem2 9942 wunex2 10160 wuncval2 10169 peano5nni 11641 1nn 11649 peano2nn 11650 om2uzrani 13321 om2uzf1oi 13322 uzrdglem 13326 uzrdgfni 13327 uzrdg0i 13328 hashkf 13693 hashgval2 13740 dftrpred2 33058 trpredpred 33067 trpredex 33076 neibastop2lem 33708 |
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