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Theorem imainss 5583
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 3234 . . . . . . . . . . 11 𝑦 ∈ V
2 vex 3234 . . . . . . . . . . 11 𝑥 ∈ V
31, 2brcnv 5337 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
4 19.8a 2090 . . . . . . . . . 10 ((𝑦𝐵𝑦𝑅𝑥) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
53, 4sylan2br 492 . . . . . . . . 9 ((𝑦𝐵𝑥𝑅𝑦) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
65ancoms 468 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝐵) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
76anim2i 592 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
8 simprl 809 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → 𝑥𝑅𝑦)
97, 8jca 553 . . . . . 6 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
109anassrs 681 . . . . 5 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11 elin 3829 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝑅𝐵)))
122elima2 5507 . . . . . . . 8 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
1312anbi2i 730 . . . . . . 7 ((𝑥𝐴𝑥 ∈ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1411, 13bitri 264 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1514anbi1i 731 . . . . 5 ((𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
1610, 15sylibr 224 . . . 4 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
1716eximi 1802 . . 3 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
181elima2 5507 . . . . 5 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
1918anbi1i 731 . . . 4 ((𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
20 elin 3829 . . . 4 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵))
21 19.41v 1917 . . . 4 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
2219, 20, 213bitr4i 292 . . 3 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
231elima2 5507 . . 3 (𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
2417, 22, 233imtr4i 281 . 2 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))))
2524ssriv 3640 1 ((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wa 383  wex 1744  wcel 2030  cin 3606  wss 3607   class class class wbr 4685  ccnv 5142  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by: (None)
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