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Mirrors > Home > MPE Home > Th. List > Mathboxes > fgraphopab | Structured version Visualization version Unicode version |
Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
fgraphopab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssxp 6060 | . . . 4 | |
2 | df-ss 3588 | . . . 4 | |
3 | 1, 2 | sylib 208 | . . 3 |
4 | ffn 6045 | . . . . 5 | |
5 | dffn5 6241 | . . . . 5 | |
6 | 4, 5 | sylib 208 | . . . 4 |
7 | 6 | ineq1d 3813 | . . 3 |
8 | 3, 7 | eqtr3d 2658 | . 2 |
9 | df-mpt 4730 | . . . 4 | |
10 | df-xp 5120 | . . . 4 | |
11 | 9, 10 | ineq12i 3812 | . . 3 |
12 | inopab 5252 | . . 3 | |
13 | anandi 871 | . . . . 5 | |
14 | ancom 466 | . . . . . . 7 | |
15 | 14 | anbi2i 730 | . . . . . 6 |
16 | anass 681 | . . . . . 6 | |
17 | eqcom 2629 | . . . . . . 7 | |
18 | 17 | anbi2i 730 | . . . . . 6 |
19 | 15, 16, 18 | 3bitr2i 288 | . . . . 5 |
20 | 13, 19 | bitr3i 266 | . . . 4 |
21 | 20 | opabbii 4717 | . . 3 |
22 | 11, 12, 21 | 3eqtri 2648 | . 2 |
23 | 8, 22 | syl6eq 2672 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cin 3573 wss 3574 copab 4712 cmpt 4729 cxp 5112 wfn 5883 wf 5884 cfv 5888 |
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-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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-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-fv 5896 |
This theorem is referenced by: fgraphxp 37789 |
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