Theorem cnvresima 6122

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 1954	. . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
2		vex 3426	. . . . . . . 8 ⊢ 𝑠 ∈ V
3	2	opelresi 5888	. . . . . . 7 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
4		vex 3426	. . . . . . . 8 ⊢ 𝑡 ∈ V
5	2, 4	opelcnv 5779	. . . . . . 7 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴))
6	2, 4	opelcnv 5779	. . . . . . . 8 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
7	6	anbi2ci 624	. . . . . . 7 ⊢ ((⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
8	3, 5, 7	3bitr4i 302	. . . . . 6 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
9	8	bianass 638	. . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
10	9	exbii 1851	. . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
11	4	elima3 5965	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹))
12	11	anbi1i 623	. . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
13	1, 10, 12	3bitr4i 302	. . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
14	4	elima3 5965	. . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)))
15		elin 3899	. . 3 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
16	13, 14, 15	3bitr4i 302	. 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
17	16	eqriv 2735	1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ∩ cin 3882  ⟨cop 4564  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by:  fimacnvinrn  6931  ramub2  16643  ramub1lem2  16656  cnrest  22344  kgencn  22615  kgencn3  22617  xkoptsub  22713  qtopres  22757  qtoprest  22776  mbfid  24704  mbfres  24713  1stpreima  30941  2ndpreima  30942  gsumhashmul  31218  cvmsss2  33136  lmhmlnmsplit  40828