Theorem imainss 6057

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 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
2		vex 3436	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5791	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
4		19.8a 2174	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
5	3, 4	sylan2br 595	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
6	5	ancoms 459	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
7	6	anim2i 617	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
8		simprl 768	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → 𝑥𝑅𝑦)
9	7, 8	jca 512	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
10	9	anassrs 468	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11		elin 3903	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)))
12	2	elima2 5975	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
13	12	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
14	11, 13	bitri 274	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
15	14	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
16	10, 15	sylibr 233	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
17	16	eximi 1837	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
18	1	elima2 5975	. . . . 5 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
19	18	anbi1i 624	. . . 4 ⊢ ((𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
20		elin 3903	. . . 4 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵))
21		19.41v 1953	. . . 4 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
22	19, 20, 21	3bitr4i 303	. . 3 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
23	1	elima2 5975	. . 3 ⊢ (𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
24	17, 22, 23	3imtr4i 292	. 2 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))))
25	24	ssriv 3925	1 ⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Syntax hints:   ∧ wa 396  ∃wex 1782   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887   class class class wbr 5074  ^◡ccnv 5588   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by: (None)