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Theorem dminss 4681
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
dminss (dom 𝑅A) ⊆ (𝑅 “ (𝑅A))

Proof of Theorem dminss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.8a 1479 . . . . . . 7 ((x A x𝑅y) → x(x A x𝑅y))
21ancoms 255 . . . . . 6 ((x𝑅y x A) → x(x A x𝑅y))
3 vex 2554 . . . . . . 7 y V
43elima2 4617 . . . . . 6 (y (𝑅A) ↔ x(x A x𝑅y))
52, 4sylibr 137 . . . . 5 ((x𝑅y x A) → y (𝑅A))
6 simpl 102 . . . . . 6 ((x𝑅y x A) → x𝑅y)
7 vex 2554 . . . . . . 7 x V
83, 7brcnv 4461 . . . . . 6 (y𝑅xx𝑅y)
96, 8sylibr 137 . . . . 5 ((x𝑅y x A) → y𝑅x)
105, 9jca 290 . . . 4 ((x𝑅y x A) → (y (𝑅A) y𝑅x))
1110eximi 1488 . . 3 (y(x𝑅y x A) → y(y (𝑅A) y𝑅x))
127eldm 4475 . . . . 5 (x dom 𝑅y x𝑅y)
1312anbi1i 431 . . . 4 ((x dom 𝑅 x A) ↔ (y x𝑅y x A))
14 elin 3120 . . . 4 (x (dom 𝑅A) ↔ (x dom 𝑅 x A))
15 19.41v 1779 . . . 4 (y(x𝑅y x A) ↔ (y x𝑅y x A))
1613, 14, 153bitr4i 201 . . 3 (x (dom 𝑅A) ↔ y(x𝑅y x A))
177elima2 4617 . . 3 (x (𝑅 “ (𝑅A)) ↔ y(y (𝑅A) y𝑅x))
1811, 16, 173imtr4i 190 . 2 (x (dom 𝑅A) → x (𝑅 “ (𝑅A)))
1918ssriv 2943 1 (dom 𝑅A) ⊆ (𝑅 “ (𝑅A))
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1378   wcel 1390  cin 2910  wss 2911   class class class wbr 3755  ccnv 4287  dom cdm 4288  cima 4291
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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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