Theorem cnvresima 6192

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		19.41v 1949	. . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
2		vex 3448	. . . . . . . 8 ⊢ 𝑠 ∈ V
3	2	opelresi 5948	. . . . . . 7 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
4		vex 3448	. . . . . . . 8 ⊢ 𝑡 ∈ V
5	2, 4	opelcnv 5836	. . . . . . 7 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴))
6	2, 4	opelcnv 5836	. . . . . . . 8 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
7	6	anbi2ci 625	. . . . . . 7 ⊢ ((⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
8	3, 5, 7	3bitr4i 303	. . . . . 6 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
9	8	bianass 642	. . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
10	9	exbii 1848	. . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
11	4	elima3 6028	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹))
12	11	anbi1i 624	. . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
13	1, 10, 12	3bitr4i 303	. . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
14	4	elima3 6028	. . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)))
15		elin 3927	. . 3 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
16	13, 14, 15	3bitr4i 303	. 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
17	16	eqriv 2726	1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∩ cin 3910  ⟨cop 4591  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634

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-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  fimacnvinrn  7026  ramub2  16963  ramub1lem2  16976  cnrest  23207  kgencn  23478  kgencn3  23480  xkoptsub  23576  qtopres  23620  qtoprest  23639  mbfid  25571  mbfres  25580  1stpreima  32682  2ndpreima  32683  gsumhashmul  33046  cvmsss2  35256  lmhmlnmsplit  43071