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Theorem imainss 4834
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 2622 . . . . . . . . . . 11 𝑦 ∈ V
2 vex 2622 . . . . . . . . . . 11 𝑥 ∈ V
31, 2brcnv 4607 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
4 19.8a 1527 . . . . . . . . . 10 ((𝑦𝐵𝑦𝑅𝑥) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
53, 4sylan2br 282 . . . . . . . . 9 ((𝑦𝐵𝑥𝑅𝑦) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
65ancoms 264 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝐵) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
76anim2i 334 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
8 simprl 498 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → 𝑥𝑅𝑦)
97, 8jca 300 . . . . . 6 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
109anassrs 392 . . . . 5 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11 elin 3181 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝑅𝐵)))
122elima2 4767 . . . . . . . 8 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
1312anbi2i 445 . . . . . . 7 ((𝑥𝐴𝑥 ∈ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1411, 13bitri 182 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1514anbi1i 446 . . . . 5 ((𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
1610, 15sylibr 132 . . . 4 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
1716eximi 1536 . . 3 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
181elima2 4767 . . . . 5 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
1918anbi1i 446 . . . 4 ((𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
20 elin 3181 . . . 4 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵))
21 19.41v 1830 . . . 4 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
2219, 20, 213bitr4i 210 . . 3 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
231elima2 4767 . . 3 (𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
2417, 22, 233imtr4i 199 . 2 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))))
2524ssriv 3027 1 ((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))
Colors of variables: wff set class
Syntax hints:  wa 102  wex 1426  wcel 1438  cin 2996  wss 2997   class class class wbr 3837  ccnv 4427  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by: (None)
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