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Theorem xpinpreima2 23293
 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.)
Assertion
Ref Expression
xpinpreima2

Proof of Theorem xpinpreima2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 xpss 4795 . . . . . . 7
2 rabss2 3258 . . . . . . 7
31, 2ax-mp 8 . . . . . 6
43a1i 10 . . . . 5
5 simprl 732 . . . . . . . 8
6 simpll 730 . . . . . . . . . 10
7 simprrl 740 . . . . . . . . . 10
86, 7sseldd 3183 . . . . . . . . 9
9 simplr 731 . . . . . . . . . 10
10 simprrr 741 . . . . . . . . . 10
119, 10sseldd 3183 . . . . . . . . 9
128, 11jca 518 . . . . . . . 8
135, 12jca 518 . . . . . . 7
14 elxp7 6154 . . . . . . 7
1513, 14sylibr 203 . . . . . 6
1615rabss3d 23138 . . . . 5
174, 16eqssd 3198 . . . 4
18 xp2 6159 . . . 4
1917, 18syl6reqr 2336 . . 3
20 inrab 3442 . . 3
2119, 20syl6eqr 2335 . 2
22 f1stres 6143 . . . . 5
23 ffn 5391 . . . . 5
24 fncnvima2 5649 . . . . 5
2522, 23, 24mp2b 9 . . . 4
26 fvres 5544 . . . . . 6
2726eleq1d 2351 . . . . 5
2827rabbiia 2780 . . . 4
2925, 28eqtri 2305 . . 3
30 f2ndres 6144 . . . . 5
31 ffn 5391 . . . . 5
32 fncnvima2 5649 . . . . 5
3330, 31, 32mp2b 9 . . . 4
34 fvres 5544 . . . . . 6
3534eleq1d 2351 . . . . 5
3635rabbiia 2780 . . . 4
3733, 36eqtri 2305 . . 3
3829, 37ineq12i 3370 . 2
3921, 38syl6eqr 2335 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1625   wcel 1686  crab 2549  cvv 2790   cin 3153   wss 3154   cxp 4689  ccnv 4690   cres 4693  cima 4694   wfn 5252  wf 5253  cfv 5257  c1st 6122  c2nd 6123 This theorem is referenced by:  cnre2csqima  23297 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-1st 6124  df-2nd 6125
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