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Theorem imainss 4682
Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
imainss ((𝑅A) ∩ B) ⊆ (𝑅 “ (A ∩ (𝑅B)))

Proof of Theorem imainss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . . . . 11 y V
2 vex 2554 . . . . . . . . . . 11 x V
31, 2brcnv 4461 . . . . . . . . . 10 (y𝑅xx𝑅y)
4 19.8a 1479 . . . . . . . . . 10 ((y B y𝑅x) → y(y B y𝑅x))
53, 4sylan2br 272 . . . . . . . . 9 ((y B x𝑅y) → y(y B y𝑅x))
65ancoms 255 . . . . . . . 8 ((x𝑅y y B) → y(y B y𝑅x))
76anim2i 324 . . . . . . 7 ((x A (x𝑅y y B)) → (x A y(y B y𝑅x)))
8 simprl 483 . . . . . . 7 ((x A (x𝑅y y B)) → x𝑅y)
97, 8jca 290 . . . . . 6 ((x A (x𝑅y y B)) → ((x A y(y B y𝑅x)) x𝑅y))
109anassrs 380 . . . . 5 (((x A x𝑅y) y B) → ((x A y(y B y𝑅x)) x𝑅y))
11 elin 3120 . . . . . . 7 (x (A ∩ (𝑅B)) ↔ (x A x (𝑅B)))
122elima2 4617 . . . . . . . 8 (x (𝑅B) ↔ y(y B y𝑅x))
1312anbi2i 430 . . . . . . 7 ((x A x (𝑅B)) ↔ (x A y(y B y𝑅x)))
1411, 13bitri 173 . . . . . 6 (x (A ∩ (𝑅B)) ↔ (x A y(y B y𝑅x)))
1514anbi1i 431 . . . . 5 ((x (A ∩ (𝑅B)) x𝑅y) ↔ ((x A y(y B y𝑅x)) x𝑅y))
1610, 15sylibr 137 . . . 4 (((x A x𝑅y) y B) → (x (A ∩ (𝑅B)) x𝑅y))
1716eximi 1488 . . 3 (x((x A x𝑅y) y B) → x(x (A ∩ (𝑅B)) x𝑅y))
181elima2 4617 . . . . 5 (y (𝑅A) ↔ x(x A x𝑅y))
1918anbi1i 431 . . . 4 ((y (𝑅A) y B) ↔ (x(x A x𝑅y) y B))
20 elin 3120 . . . 4 (y ((𝑅A) ∩ B) ↔ (y (𝑅A) y B))
21 19.41v 1779 . . . 4 (x((x A x𝑅y) y B) ↔ (x(x A x𝑅y) y B))
2219, 20, 213bitr4i 201 . . 3 (y ((𝑅A) ∩ B) ↔ x((x A x𝑅y) y B))
231elima2 4617 . . 3 (y (𝑅 “ (A ∩ (𝑅B))) ↔ x(x (A ∩ (𝑅B)) x𝑅y))
2417, 22, 233imtr4i 190 . 2 (y ((𝑅A) ∩ B) → y (𝑅 “ (A ∩ (𝑅B))))
2524ssriv 2943 1 ((𝑅A) ∩ B) ⊆ (𝑅 “ (A ∩ (𝑅B)))
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  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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