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	⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Proof of Theorem imainss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3461	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
2		vex 3461	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5870	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
4		19.8a 2220	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
5	3, 4	sylan2br 607	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
6	5	ancoms 464	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
7	6	anim2i 629	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
8		simprl 783	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → 𝑥𝑅𝑦)
9	7, 8	jca 521	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
10	9	anassrs 473	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11		elin 3922	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)))
12	2	elima2 6070	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
13	12	anbi2i 635	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
14	11, 13	bitri 278	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
15	14	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
16	10, 15	sylibr 237	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
17	16	eximi 1868	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
18	1	elima2 6070	. . . . 5 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
19	18	anbi1i 636	. . . 4 ⊢ ((𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
20		elin 3922	. . . 4 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵))
21		19.41v 1982	. . . 4 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
22	19, 20, 21	3bitr4i 306	. . 3 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
23	1	elima2 6070	. . 3 ⊢ (𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
24	17, 22, 23	3imtr4i 295	. 2 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))))
25	24	ssriv 3942	1 ⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2146   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111  ^◡ccnv 5662   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by: (None)