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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 2137 | . . 3 ⊢ dom 𝐴 = dom 𝐴 | |
2 | 1 | biantru 300 | . 2 ⊢ (Fun 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴)) |
3 | df-fn 5121 | . 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 1331 dom cdm 4534 Fun wfun 5112 Fn wfn 5113 |
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 1425 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-cleq 2130 df-fn 5121 |
This theorem is referenced by: funfnd 5149 funssxp 5287 funforn 5347 funbrfvb 5457 funopfvb 5458 ssimaex 5475 fvco 5484 eqfunfv 5516 fvimacnvi 5527 unpreima 5538 respreima 5541 elrnrexdm 5552 elrnrexdmb 5553 ffvresb 5576 funresdfunsnss 5616 resfunexg 5634 funex 5636 elunirn 5660 smores 6182 smores2 6184 tfrlem1 6198 funresdfunsndc 6395 fundmfibi 6820 resunimafz0 10567 fclim 11056 |
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