Theorem imainss 5043

Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by set.mm contributors, 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		simpll 730	. . . . . 6 ⊢ (((x ∈ A ∧ xRy) ∧ y ∈ B) → x ∈ A)
2		brcnv 4893	. . . . . . . . 9 ⊢ (y◡Rx ↔ xRy)
3		19.8a 1756	. . . . . . . . 9 ⊢ ((y ∈ B ∧ y◡Rx) → ∃y(y ∈ B ∧ y◡Rx))
4	2, 3	sylan2br 462	. . . . . . . 8 ⊢ ((y ∈ B ∧ xRy) → ∃y(y ∈ B ∧ y◡Rx))
5	4	ancoms 439	. . . . . . 7 ⊢ ((xRy ∧ y ∈ B) → ∃y(y ∈ B ∧ y◡Rx))
6	5	adantll 694	. . . . . 6 ⊢ (((x ∈ A ∧ xRy) ∧ y ∈ B) → ∃y(y ∈ B ∧ y◡Rx))
7	1, 6	jca 518	. . . . 5 ⊢ (((x ∈ A ∧ xRy) ∧ y ∈ B) → (x ∈ A ∧ ∃y(y ∈ B ∧ y◡Rx)))
8		simplr 731	. . . . 5 ⊢ (((x ∈ A ∧ xRy) ∧ y ∈ B) → xRy)
9		elin 3220	. . . . . . 7 ⊢ (x ∈ (A ∩ (◡R “ B)) ↔ (x ∈ A ∧ x ∈ (◡R “ B)))
10		elima2 4756	. . . . . . . 8 ⊢ (x ∈ (◡R “ B) ↔ ∃y(y ∈ B ∧ y◡Rx))
11	10	anbi2i 675	. . . . . . 7 ⊢ ((x ∈ A ∧ x ∈ (◡R “ B)) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ y◡Rx)))
12	9, 11	bitri 240	. . . . . 6 ⊢ (x ∈ (A ∩ (◡R “ B)) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ y◡Rx)))
13	12	anbi1i 676	. . . . 5 ⊢ ((x ∈ (A ∩ (◡R “ B)) ∧ xRy) ↔ ((x ∈ A ∧ ∃y(y ∈ B ∧ y◡Rx)) ∧ xRy))
14	7, 8, 13	sylanbrc 645	. . . 4 ⊢ (((x ∈ A ∧ xRy) ∧ y ∈ B) → (x ∈ (A ∩ (◡R “ B)) ∧ xRy))
15	14	eximi 1576	. . 3 ⊢ (∃x((x ∈ A ∧ xRy) ∧ y ∈ B) → ∃x(x ∈ (A ∩ (◡R “ B)) ∧ xRy))
16		elima2 4756	. . . . 5 ⊢ (y ∈ (R “ A) ↔ ∃x(x ∈ A ∧ xRy))
17	16	anbi1i 676	. . . 4 ⊢ ((y ∈ (R “ A) ∧ y ∈ B) ↔ (∃x(x ∈ A ∧ xRy) ∧ y ∈ B))
18		elin 3220	. . . 4 ⊢ (y ∈ ((R “ A) ∩ B) ↔ (y ∈ (R “ A) ∧ y ∈ B))
19		19.41v 1901	. . . 4 ⊢ (∃x((x ∈ A ∧ xRy) ∧ y ∈ B) ↔ (∃x(x ∈ A ∧ xRy) ∧ y ∈ B))
20	17, 18, 19	3bitr4i 268	. . 3 ⊢ (y ∈ ((R “ A) ∩ B) ↔ ∃x((x ∈ A ∧ xRy) ∧ y ∈ B))
21		elima2 4756	. . 3 ⊢ (y ∈ (R “ (A ∩ (◡R “ B))) ↔ ∃x(x ∈ (A ∩ (◡R “ B)) ∧ xRy))
22	15, 20, 21	3imtr4i 257	. 2 ⊢ (y ∈ ((R “ A) ∩ B) → y ∈ (R “ (A ∩ (◡R “ B))))
23	22	ssriv 3278	1 ⊢ ((R “ A) ∩ B) ⊆ (R “ (A ∩ (◡R “ B)))

Syntax hints:   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ∩ cin 3209   ⊆ wss 3258   class class class wbr 4640   “ cima 4723  ^◡ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-cnv 4786

This theorem is referenced by: (None)