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Theorem imainss 6153
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 3478 . . . . . . . . . . 11 𝑦 ∈ V
2 vex 3478 . . . . . . . . . . 11 𝑥 ∈ V
31, 2brcnv 5882 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
4 19.8a 2174 . . . . . . . . . 10 ((𝑦𝐵𝑦𝑅𝑥) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
53, 4sylan2br 595 . . . . . . . . 9 ((𝑦𝐵𝑥𝑅𝑦) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
65ancoms 459 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝐵) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
76anim2i 617 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
8 simprl 769 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → 𝑥𝑅𝑦)
97, 8jca 512 . . . . . 6 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
109anassrs 468 . . . . 5 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11 elin 3964 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝑅𝐵)))
122elima2 6065 . . . . . . . 8 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
1312anbi2i 623 . . . . . . 7 ((𝑥𝐴𝑥 ∈ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1411, 13bitri 274 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1514anbi1i 624 . . . . 5 ((𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
1610, 15sylibr 233 . . . 4 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
1716eximi 1837 . . 3 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
181elima2 6065 . . . . 5 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
1918anbi1i 624 . . . 4 ((𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
20 elin 3964 . . . 4 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵))
21 19.41v 1953 . . . 4 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
2219, 20, 213bitr4i 302 . . 3 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
231elima2 6065 . . 3 (𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
2417, 22, 233imtr4i 291 . 2 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))))
2524ssriv 3986 1 ((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wa 396  wex 1781  wcel 2106  cin 3947  wss 3948   class class class wbr 5148  ccnv 5675  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by: (None)
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