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Theorem relrelss 4787
Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
relrelss ((Rel A Rel dom A) ↔ A ⊆ ((V × V) × V))

Proof of Theorem relrelss
StepHypRef Expression
1 df-rel 4295 . . 3 (Rel dom A ↔ dom A ⊆ (V × V))
21anbi2i 430 . 2 ((Rel A Rel dom A) ↔ (Rel A dom A ⊆ (V × V)))
3 relssdmrn 4784 . . . 4 (Rel AA ⊆ (dom A × ran A))
4 ssv 2959 . . . . 5 ran A ⊆ V
5 xpss12 4388 . . . . 5 ((dom A ⊆ (V × V) ran A ⊆ V) → (dom A × ran A) ⊆ ((V × V) × V))
64, 5mpan2 401 . . . 4 (dom A ⊆ (V × V) → (dom A × ran A) ⊆ ((V × V) × V))
73, 6sylan9ss 2952 . . 3 ((Rel A dom A ⊆ (V × V)) → A ⊆ ((V × V) × V))
8 xpss 4389 . . . . . 6 ((V × V) × V) ⊆ (V × V)
9 sstr 2947 . . . . . 6 ((A ⊆ ((V × V) × V) ((V × V) × V) ⊆ (V × V)) → A ⊆ (V × V))
108, 9mpan2 401 . . . . 5 (A ⊆ ((V × V) × V) → A ⊆ (V × V))
11 df-rel 4295 . . . . 5 (Rel AA ⊆ (V × V))
1210, 11sylibr 137 . . . 4 (A ⊆ ((V × V) × V) → Rel A)
13 dmss 4477 . . . . 5 (A ⊆ ((V × V) × V) → dom A ⊆ dom ((V × V) × V))
14 vn0m 3226 . . . . . 6 x x V
15 dmxpm 4498 . . . . . 6 (x x V → dom ((V × V) × V) = (V × V))
1614, 15ax-mp 7 . . . . 5 dom ((V × V) × V) = (V × V)
1713, 16syl6sseq 2985 . . . 4 (A ⊆ ((V × V) × V) → dom A ⊆ (V × V))
1812, 17jca 290 . . 3 (A ⊆ ((V × V) × V) → (Rel A dom A ⊆ (V × V)))
197, 18impbii 117 . 2 ((Rel A dom A ⊆ (V × V)) ↔ A ⊆ ((V × V) × V))
202, 19bitri 173 1 ((Rel A Rel dom A) ↔ A ⊆ ((V × V) × V))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  dftpos3  5818  tpostpos2  5821
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