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Mirrors > Home > ILE Home > Th. List > funfn | GIF version |
Description: An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
funfn | ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2170 | . . 3 ⊢ dom 𝐴 = dom 𝐴 | |
2 | 1 | biantru 300 | . 2 ⊢ (Fun 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) |
3 | df-fn 5199 | . 2 ⊢ (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) | |
4 | 2, 3 | bitr4i 186 | 1 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1348 dom cdm 4609 Fun wfun 5190 Fn wfn 5191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1442 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 df-fn 5199 |
This theorem is referenced by: funfnd 5227 funssxp 5365 funforn 5425 funbrfvb 5537 funopfvb 5538 ssimaex 5555 fvco 5564 eqfunfv 5596 fvimacnvi 5608 unpreima 5619 respreima 5622 elrnrexdm 5633 elrnrexdmb 5634 ffvresb 5657 funresdfunsnss 5697 resfunexg 5715 funex 5717 elunirn 5743 smores 6269 smores2 6271 tfrlem1 6285 funresdfunsndc 6483 fundmfibi 6913 resunimafz0 10755 fclim 11246 |
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