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Theorem imainss 6152
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 3476 . . . . . . . . . . 11 𝑦 ∈ V
2 vex 3476 . . . . . . . . . . 11 𝑥 ∈ V
31, 2brcnv 5881 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
4 19.8a 2172 . . . . . . . . . 10 ((𝑦𝐵𝑦𝑅𝑥) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
53, 4sylan2br 593 . . . . . . . . 9 ((𝑦𝐵𝑥𝑅𝑦) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
65ancoms 457 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝐵) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
76anim2i 615 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
8 simprl 767 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → 𝑥𝑅𝑦)
97, 8jca 510 . . . . . 6 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
109anassrs 466 . . . . 5 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11 elin 3963 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝑅𝐵)))
122elima2 6064 . . . . . . . 8 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
1312anbi2i 621 . . . . . . 7 ((𝑥𝐴𝑥 ∈ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1411, 13bitri 274 . . . . . 6 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
1514anbi1i 622 . . . . 5 ((𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
1610, 15sylibr 233 . . . 4 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
1716eximi 1835 . . 3 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
181elima2 6064 . . . . 5 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
1918anbi1i 622 . . . 4 ((𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
20 elin 3963 . . . 4 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵))
21 19.41v 1951 . . . 4 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
2219, 20, 213bitr4i 302 . . 3 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
231elima2 6064 . . 3 (𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
2417, 22, 233imtr4i 291 . 2 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))))
2524ssriv 3985 1 ((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wa 394  wex 1779  wcel 2104  cin 3946  wss 3947   class class class wbr 5147  ccnv 5674  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by: (None)
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