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Mirrors > Home > NFE Home > Th. List > funfn | GIF version |
Description: An equivalence for the function predicate. (Contributed by set.mm contributors, 13-Aug-2004.) |
Ref | Expression |
---|---|
funfn | ⊢ (Fun A ↔ A Fn dom A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2353 | . . 3 ⊢ dom A = dom A | |
2 | 1 | biantru 491 | . 2 ⊢ (Fun A ↔ (Fun A ∧ dom A = dom A)) |
3 | df-fn 4790 | . 2 ⊢ (A Fn dom A ↔ (Fun A ∧ dom A = dom A)) | |
4 | 2, 3 | bitr4i 243 | 1 ⊢ (Fun A ↔ A Fn dom A) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 = wceq 1642 dom cdm 4772 Fun wfun 4775 Fn wfn 4776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-cleq 2346 df-fn 4790 |
This theorem is referenced by: funssxp 5233 f1funfun 5263 funforn 5276 funbrfvb 5360 fvco 5383 eqfunfv 5397 fvimacnvi 5402 unpreima 5408 inpreima 5409 respreima 5410 ffvresb 5431 fnfullfun 5858 fvfullfun 5864 sbthlem1 6203 sbthlem3 6205 |
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