Theorem cnvresima 5078

Description: An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Assertion

Ref	Expression
cnvresima	⊢ (◡(F ↾ A) “ B) = ((◡F “ B) ∩ A)

Proof of Theorem cnvresima

Dummy variables t s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elima3 4757	. . 3 ⊢ (t ∈ (◡(F ↾ A) “ B) ↔ ∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡(F ↾ A)))
2		anass 630	. . . . . 6 ⊢ (((s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A) ↔ (s ∈ B ∧ (⟨s, t⟩ ∈ ◡F ∧ t ∈ A)))
3		opelres 4951	. . . . . . . 8 ⊢ (⟨t, s⟩ ∈ (F ↾ A) ↔ (⟨t, s⟩ ∈ F ∧ t ∈ A))
4		opelcnv 4894	. . . . . . . 8 ⊢ (⟨s, t⟩ ∈ ◡(F ↾ A) ↔ ⟨t, s⟩ ∈ (F ↾ A))
5		opelcnv 4894	. . . . . . . . 9 ⊢ (⟨s, t⟩ ∈ ◡F ↔ ⟨t, s⟩ ∈ F)
6	5	anbi1i 676	. . . . . . . 8 ⊢ ((⟨s, t⟩ ∈ ◡F ∧ t ∈ A) ↔ (⟨t, s⟩ ∈ F ∧ t ∈ A))
7	3, 4, 6	3bitr4ri 269	. . . . . . 7 ⊢ ((⟨s, t⟩ ∈ ◡F ∧ t ∈ A) ↔ ⟨s, t⟩ ∈ ◡(F ↾ A))
8	7	anbi2i 675	. . . . . 6 ⊢ ((s ∈ B ∧ (⟨s, t⟩ ∈ ◡F ∧ t ∈ A)) ↔ (s ∈ B ∧ ⟨s, t⟩ ∈ ◡(F ↾ A)))
9	2, 8	bitr2i 241	. . . . 5 ⊢ ((s ∈ B ∧ ⟨s, t⟩ ∈ ◡(F ↾ A)) ↔ ((s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A))
10	9	exbii 1582	. . . 4 ⊢ (∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡(F ↾ A)) ↔ ∃s((s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A))
11		19.41v 1901	. . . 4 ⊢ (∃s((s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A) ↔ (∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A))
12	10, 11	bitri 240	. . 3 ⊢ (∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡(F ↾ A)) ↔ (∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A))
13		elin 3220	. . . 4 ⊢ (t ∈ ((◡F “ B) ∩ A) ↔ (t ∈ (◡F “ B) ∧ t ∈ A))
14		elima3 4757	. . . . 5 ⊢ (t ∈ (◡F “ B) ↔ ∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡F))
15	14	anbi1i 676	. . . 4 ⊢ ((t ∈ (◡F “ B) ∧ t ∈ A) ↔ (∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A))
16	13, 15	bitr2i 241	. . 3 ⊢ ((∃s(s ∈ B ∧ ⟨s, t⟩ ∈ ◡F) ∧ t ∈ A) ↔ t ∈ ((◡F “ B) ∩ A))
17	1, 12, 16	3bitri 262	. 2 ⊢ (t ∈ (◡(F ↾ A) “ B) ↔ t ∈ ((◡F “ B) ∩ A))
18	17	eqriv 2350	1 ⊢ (◡(F ↾ A) “ B) = ((◡F “ B) ∩ A)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ⟨cop 4562   “ cima 4723  ^◡ccnv 4772   ↾ cres 4775

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-xp 4785  df-cnv 4786  df-res 4789

This theorem is referenced by: (None)