Theorem imainss 5019

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 2729	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
2		vex 2729	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 4787	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
4		19.8a 1578	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
5	3, 4	sylan2br 286	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
6	5	ancoms 266	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
7	6	anim2i 340	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
8		simprl 521	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → 𝑥𝑅𝑦)
9	7, 8	jca 304	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
10	9	anassrs 398	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11		elin 3305	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)))
12	2	elima2 4952	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
13	12	anbi2i 453	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
14	11, 13	bitri 183	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
15	14	anbi1i 454	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
16	10, 15	sylibr 133	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
17	16	eximi 1588	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
18	1	elima2 4952	. . . . 5 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
19	18	anbi1i 454	. . . 4 ⊢ ((𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
20		elin 3305	. . . 4 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵))
21		19.41v 1890	. . . 4 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
22	19, 20, 21	3bitr4i 211	. . 3 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
23	1	elima2 4952	. . 3 ⊢ (𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
24	17, 22, 23	3imtr4i 200	. 2 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))))
25	24	ssriv 3146	1 ⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Syntax hints:   ∧ wa 103  ∃wex 1480   ∈ wcel 2136   ∩ cin 3115   ⊆ wss 3116   class class class wbr 3982  ^◡ccnv 4603   “ cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by: (None)