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Theorem imainss 5546
 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 3201 . . . . . . . . . . 11 𝑦 ∈ V
2 vex 3201 . . . . . . . . . . 11 𝑥 ∈ V
31, 2brcnv 5303 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
4 19.8a 2051 . . . . . . . . . 10 ((𝑦𝐵𝑦𝑅𝑥) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
53, 4sylan2br 493 . . . . . . . . 9 ((𝑦𝐵𝑥𝑅𝑦) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
65ancoms 469 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝐵) → ∃𝑦(𝑦𝐵𝑦𝑅𝑥))
76anim2i 593 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)))
8 simprl 794 . . . . . . 7 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → 𝑥𝑅𝑦)
97, 8jca 554 . . . . . 6 ((𝑥𝐴 ∧ (𝑥𝑅𝑦𝑦𝐵)) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
109anassrs 680 . . . . 5 (((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ((𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝑦𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11 elin 3794 . . . . . . 7 (𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ↔ (𝑥𝐴𝑥 ∈ (𝑅𝐵)))
122elima2 5470 . . . . . . . 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 1761 . . 3 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
181elima2 5470 . . . . 5 (𝑦 ∈ (𝑅𝐴) ↔ ∃𝑥(𝑥𝐴𝑥𝑅𝑦))
1918anbi1i 731 . . . 4 ((𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
20 elin 3794 . . . 4 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅𝐴) ∧ 𝑦𝐵))
21 19.41v 1913 . . . 4 (∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵) ↔ (∃𝑥(𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
2219, 20, 213bitr4i 292 . . 3 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥𝐴𝑥𝑅𝑦) ∧ 𝑦𝐵))
231elima2 5470 . . 3 (𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (𝑅𝐵)) ∧ 𝑥𝑅𝑦))
2417, 22, 233imtr4i 281 . 2 (𝑦 ∈ ((𝑅𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (𝑅𝐵))))
2524ssriv 3605 1 ((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384  ∃wex 1703   ∈ wcel 1989   ∩ cin 3571   ⊆ wss 3572   class class class wbr 4651  ◡ccnv 5111   “ cima 5115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125 This theorem is referenced by: (None)
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