Theorem fcnvres 5244

Description: The converse of a restriction of a function. (Contributed by set.mm contributors, 26-Mar-1998.)

Assertion

Ref	Expression
fcnvres	⊢ (F:A–→B → ◡(F ↾ A) = (◡F ↾ B))

Proof of Theorem fcnvres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4641	. . . . 5 ⊢ (xFy ↔ ⟨x, y⟩ ∈ F)
2		ffn 5224	. . . . . . 7 ⊢ (F:A–→B → F Fn A)
3		fnbr 5186	. . . . . . 7 ⊢ ((F Fn A ∧ xFy) → x ∈ A)
4	2, 3	sylan 457	. . . . . 6 ⊢ ((F:A–→B ∧ xFy) → x ∈ A)
5		frn 5229	. . . . . . 7 ⊢ (F:A–→B → ran F ⊆ B)
6		brelrn 4961	. . . . . . 7 ⊢ (xFy → y ∈ ran F)
7		ssel2 3269	. . . . . . 7 ⊢ ((ran F ⊆ B ∧ y ∈ ran F) → y ∈ B)
8	5, 6, 7	syl2an 463	. . . . . 6 ⊢ ((F:A–→B ∧ xFy) → y ∈ B)
9	4, 8	2thd 231	. . . . 5 ⊢ ((F:A–→B ∧ xFy) → (x ∈ A ↔ y ∈ B))
10	1, 9	sylan2br 462	. . . 4 ⊢ ((F:A–→B ∧ ⟨x, y⟩ ∈ F) → (x ∈ A ↔ y ∈ B))
11	10	pm5.32da 622	. . 3 ⊢ (F:A–→B → ((⟨x, y⟩ ∈ F ∧ x ∈ A) ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B)))
12		opelcnv 4894	. . . 4 ⊢ (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨x, y⟩ ∈ (F ↾ A))
13		opelres 4951	. . . 4 ⊢ (⟨x, y⟩ ∈ (F ↾ A) ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A))
14	12, 13	bitri 240	. . 3 ⊢ (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ (⟨x, y⟩ ∈ F ∧ x ∈ A))
15		opelres 4951	. . . 4 ⊢ (⟨y, x⟩ ∈ (◡F ↾ B) ↔ (⟨y, x⟩ ∈ ◡F ∧ y ∈ B))
16		opelcnv 4894	. . . . 5 ⊢ (⟨y, x⟩ ∈ ◡F ↔ ⟨x, y⟩ ∈ F)
17	16	anbi1i 676	. . . 4 ⊢ ((⟨y, x⟩ ∈ ◡F ∧ y ∈ B) ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B))
18	15, 17	bitri 240	. . 3 ⊢ (⟨y, x⟩ ∈ (◡F ↾ B) ↔ (⟨x, y⟩ ∈ F ∧ y ∈ B))
19	11, 14, 18	3bitr4g 279	. 2 ⊢ (F:A–→B → (⟨y, x⟩ ∈ ◡(F ↾ A) ↔ ⟨y, x⟩ ∈ (◡F ↾ B)))
20	19	eqrelrdv 4853	1 ⊢ (F:A–→B → ◡(F ↾ A) = (◡F ↾ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562   class class class wbr 4640  ^◡ccnv 4772  ran crn 4774   ↾ cres 4775   Fn wfn 4777  –→wf 4778

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-rn 4787  df-dm 4788  df-res 4789  df-fn 4791  df-f 4792

This theorem is referenced by: (None)