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Mirrors > Home > MPE Home > Th. List > dffun6 | Structured version Visualization version GIF version |
Description: Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
Ref | Expression |
---|---|
dffun6 | ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝐹 | |
3 | 1, 2 | dffun6f 5818 | 1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃*wmo 2459 class class class wbr 4583 Rel wrel 5043 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funmo 5820 dffun7 5830 fununfun 5848 funcnvsn 5850 funcnv2 5871 svrelfun 5875 fnres 5921 nfunsn 6135 dff3 6280 brdom3 9231 nqerf 9631 shftfn 13661 cnextfun 21678 perfdvf 23473 taylf 23919 |
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