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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 2975 | . 2 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2975 | . 2 ⊢ Ⅎ𝑦𝐹 | |
3 | 1, 2 | dffun6f 6362 | 1 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1529 ∃*wmo 2614 class class class wbr 5057 Rel wrel 5553 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-id 5453 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: funmo 6364 dffun7 6375 fununfun 6395 funcnvsn 6397 funcnv2 6415 svrelfun 6419 fnres 6467 nfunsn 6700 dff3 6859 brdom3 9942 nqerf 10344 shftfn 14424 cnextfun 22664 perfdvf 24493 taylf 24941 funressnvmo 43271 |
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