Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpinpreima | Structured version Visualization version Unicode version |
Description: Rewrite the cartesian product of two sets as the intersection of their preimage by and , the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
Ref | Expression |
---|---|
xpinpreima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inrab 3899 | . 2 | |
2 | f1stres 7190 | . . . . 5 | |
3 | ffn 6045 | . . . . 5 | |
4 | fncnvima2 6339 | . . . . 5 | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 |
6 | fvres 6207 | . . . . . 6 | |
7 | 6 | eleq1d 2686 | . . . . 5 |
8 | 7 | rabbiia 3185 | . . . 4 |
9 | 5, 8 | eqtri 2644 | . . 3 |
10 | f2ndres 7191 | . . . . 5 | |
11 | ffn 6045 | . . . . 5 | |
12 | fncnvima2 6339 | . . . . 5 | |
13 | 10, 11, 12 | mp2b 10 | . . . 4 |
14 | fvres 6207 | . . . . . 6 | |
15 | 14 | eleq1d 2686 | . . . . 5 |
16 | 15 | rabbiia 3185 | . . . 4 |
17 | 13, 16 | eqtri 2644 | . . 3 |
18 | 9, 17 | ineq12i 3812 | . 2 |
19 | xp2 7203 | . 2 | |
20 | 1, 18, 19 | 3eqtr4ri 2655 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 cin 3573 cxp 5112 ccnv 5113 cres 5116 cima 5117 wfn 5883 wf 5884 cfv 5888 c1st 7166 c2nd 7167 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-ne 2795 df-ral 2917 df-rex 2918 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-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: (None) |
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