Theorem imainss 6127

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 3451	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
2		vex 3451	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5846	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
4		19.8a 2182	. . . . . . . . . 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 458	. . . . . . . 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 770	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → 𝑥𝑅𝑦)
9	7, 8	jca 511	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
10	9	anassrs 467	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11		elin 3930	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)))
12	2	elima2 6037	. . . . . . . 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 275	. . . . . 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 234	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
17	16	eximi 1835	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
18	1	elima2 6037	. . . . 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 3930	. . . 4 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵))
21		19.41v 1949	. . . 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 6037	. . 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 3950	1 ⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107  ^◡ccnv 5637   “ cima 5641

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by: (None)