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Mirrors > Home > ILE Home > Th. List > frec2uz0d | Unicode version |
Description: The mapping is a one-to-one mapping from onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number (normally 0 for the upper integers or 1 for the upper integers ), 1 maps to + 1, etc. This theorem shows the value of at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
Ref | Expression |
---|---|
frec2uz0d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.2 | . . 3 frec | |
2 | 1 | fveq1i 5179 | . 2 frec |
3 | frec2uz.1 | . . 3 | |
4 | frec0g 5983 | . . 3 frec | |
5 | 3, 4 | syl 14 | . 2 frec |
6 | 2, 5 | syl5eq 2084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wcel 1393 c0 3224 cmpt 3818 cfv 4902 (class class class)co 5512 freccfrec 5977 c1 6890 caddc 6892 cz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 df-frec 5978 |
This theorem is referenced by: frec2uzzd 9186 frec2uzuzd 9188 frec2uzrand 9191 frec2uzrdg 9195 frecfzennn 9203 |
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