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Mirrors > Home > MPE Home > Th. List > funfn | Structured version Visualization version GIF version |
Description: A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funfn | ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ dom 𝐴 = dom 𝐴 | |
2 | 1 | biantru 529 | . 2 ⊢ (Fun 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) |
3 | df-fn 6565 | . 2 ⊢ (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 dom cdm 5688 Fun wfun 6556 Fn wfn 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-fn 6565 |
This theorem is referenced by: funfnd 6598 funssxp 6764 funforn 6827 funbrfvb 6961 funopfvb 6962 ssimaex 6993 fvco 7006 fvco4i 7009 eqfunfv 7055 fvimacnvi 7071 unpreima 7082 respreima 7085 elrnrexdm 7108 elrnrexdmb 7109 ffvresb 7144 funiun 7166 funressn 7178 funresdfunsn 7208 funex 7238 elunirn 7270 suppval1 8189 funsssuppss 8213 smores 8390 rdgsucg 8461 rdglimg 8463 imafiOLD 9351 fundmfibi 9373 residfi 9375 mptfi 9388 ordtypelem6 9560 ordtypelem7 9561 harwdom 9628 ackbij2 10279 mptct 10575 smobeth 10623 hashkf 14367 hashfun 14472 fclim 15585 coapm 18124 psgnghm 21615 lindfrn 21858 elno3 27714 noextenddif 27727 noextendlt 27728 noextendgt 27729 nosupbnd2lem1 27774 noetasuplem4 27795 ausgrumgri 29198 dfnbgr3 29369 wlkiswwlks1 29896 vdn0conngrumgrv2 30224 hlimf 31265 adj1o 31922 abrexdomjm 32534 iunpreima 32584 fresf1o 32647 unipreima 32659 xppreima 32661 suppiniseg 32700 fdifsuppconst 32703 ressupprn 32704 mptctf 32734 orvcval2 34439 fineqvac 35089 fullfunfnv 35927 fullfunfv 35928 abrexdom 37716 diaf11N 41031 dibf11N 41143 imadomfi 41983 gneispace3 44122 fresfo 46997 funbrafvb 47105 funopafvb 47106 funbrafv22b 47199 funopafv2b 47200 dfclnbgr3 47750 grimuhgr 47815 |
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