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Mirrors > Home > MPE Home > Th. List > Mathboxes > nofun | Structured version Visualization version GIF version |
Description: A surreal is a function. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
nofun | ⊢ (𝐴 ∈ No → Fun 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elno 33266 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | ffun 6490 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → Fun 𝐴) | |
3 | 2 | rexlimivw 3241 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → Fun 𝐴) |
4 | 1, 3 | sylbi 220 | 1 ⊢ (𝐴 ∈ No → Fun 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∃wrex 3107 {cpr 4527 Oncon0 6159 Fun wfun 6318 ⟶wf 6320 1oc1o 8078 2oc2o 8079 No csur 33260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-no 33263 |
This theorem is referenced by: nofnbday 33272 elno2 33274 nofv 33277 sltres 33282 nosepon 33285 noextend 33286 noextendseq 33287 noextenddif 33288 noextendlt 33289 noextendgt 33290 nolesgn2ores 33292 nosepssdm 33303 nolt02olem 33311 nolt02o 33312 nosupno 33316 nosupres 33320 nosupbnd1lem5 33325 nosupbnd1 33327 nosupbnd2lem1 33328 nosupbnd2 33329 noetalem2 33331 noetalem3 33332 |
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