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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 2890 | . 2 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2890 | . 2 ⊢ Ⅎ𝑦𝐹 | |
3 | 1, 2 | dffun6f 6051 | 1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∀wal 1618 ∃*wmo 2596 class class class wbr 4792 Rel wrel 5259 Fun wfun 6031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pr 5043 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rab 3047 df-v 3330 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-op 4316 df-br 4793 df-opab 4853 df-id 5162 df-cnv 5262 df-co 5263 df-fun 6039 |
This theorem is referenced by: funmo 6053 dffun7 6064 fununfun 6083 funcnvsn 6085 funcnv2 6106 svrelfun 6110 fnres 6156 nfunsn 6374 dff3 6523 brdom3 9513 nqerf 9915 shftfn 13983 cnextfun 22040 perfdvf 23837 taylf 24285 |
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