![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 32388 | . 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o}) | |
2 | ffun 6294 | . . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → Fun 𝐴) | |
3 | 2 | rexlimivw 3211 | . 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → Fun 𝐴) |
4 | 1, 3 | sylbi 209 | 1 ⊢ (𝐴 ∈ No → Fun 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∃wrex 3091 {cpr 4400 Oncon0 5976 Fun wfun 6129 ⟶wf 6131 1oc1o 7836 2oc2o 7837 No csur 32382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-no 32385 |
This theorem is referenced by: nofnbday 32394 elno2 32396 nofv 32399 sltres 32404 nosepon 32407 noextend 32408 noextendseq 32409 noextenddif 32410 noextendlt 32411 noextendgt 32412 nolesgn2ores 32414 nosepssdm 32425 nolt02olem 32433 nolt02o 32434 nosupno 32438 nosupres 32442 nosupbnd1lem5 32447 nosupbnd1 32449 nosupbnd2lem1 32450 nosupbnd2 32451 noetalem2 32453 noetalem3 32454 |
Copyright terms: Public domain | W3C validator |