Theorem cnvresima 5226

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		vex 2805	. . . 4 ⊢ 𝑡 ∈ V
2	1	elima3 5083	. . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)))
3	1	elima3 5083	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹))
4	3	anbi1i 458	. . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
5		elin 3390	. . . 4 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
6		vex 2805	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
7	6, 1	opelcnv 4912	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴))
8	6	opelres 5018	. . . . . . . . . 10 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹 ∧ 𝑡 ∈ 𝐴))
9	6, 1	opelcnv 4912	. . . . . . . . . . 11 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
10	9	anbi1i 458	. . . . . . . . . 10 ⊢ ((⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹 ∧ 𝑡 ∈ 𝐴))
11	8, 10	bitr4i 187	. . . . . . . . 9 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
12	7, 11	bitri 184	. . . . . . . 8 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
13	12	anbi2i 457	. . . . . . 7 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑠 ∈ 𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)))
14		anass 401	. . . . . . 7 ⊢ (((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (𝑠 ∈ 𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)))
15	13, 14	bitr4i 187	. . . . . 6 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
16	15	exbii 1653	. . . . 5 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
17		19.41v 1951	. . . . 5 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
18	16, 17	bitri 184	. . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
19	4, 5, 18	3bitr4ri 213	. . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
20	2, 19	bitri 184	. 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
21	20	eqriv 2228	1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ∩ cin 3199  ⟨cop 3672  ^◡ccnv 4724   ↾ cres 4727   “ cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  cnrest  14958