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Mirrors > Home > MPE Home > Th. List > svrelfun | Structured version Visualization version GIF version |
Description: A single-valued relation is a function. (See fun2cnv 6427 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
svrelfun | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 6372 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | |
2 | fun2cnv 6427 | . . 3 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) | |
3 | 2 | anbi2i 624 | . 2 ⊢ ((Rel 𝐴 ∧ Fun ◡◡𝐴) ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃*wmo 2620 class class class wbr 5068 ◡ccnv 5556 Rel wrel 5562 Fun wfun 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-fun 6359 |
This theorem is referenced by: (None) |
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