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Mirrors > Home > MPE Home > Th. List > funop | Structured version Visualization version Unicode version |
Description: An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) |
Ref | Expression |
---|---|
funopsn.x | |
funopsn.y |
Ref | Expression |
---|---|
funop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | funopsn.x | . . . 4 | |
3 | funopsn.y | . . . 4 | |
4 | 2, 3 | funopsn 6413 | . . 3 |
5 | 1, 4 | mpan2 707 | . 2 |
6 | vex 3203 | . . . . . 6 | |
7 | 6, 6 | funsn 5939 | . . . . 5 |
8 | funeq 5908 | . . . . 5 | |
9 | 7, 8 | mpbiri 248 | . . . 4 |
10 | 9 | adantl 482 | . . 3 |
11 | 10 | exlimiv 1858 | . 2 |
12 | 5, 11 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cvv 3200 csn 4177 cop 4183 wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: funopdmsn 6415 funsndifnop 6416 |
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