Theorem cnvresima 4810

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 2560	. . . 4 ⊢ 𝑡 ∈ V
2	1	elima3 4675	. . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)))
3	1	elima3 4675	. . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹))
4	3	anbi1i 431	. . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
5		elin 3126	. . . 4 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
6		vex 2560	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
7	6, 1	opelcnv 4517	. . . . . . . . 9 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴))
8	6	opelres 4617	. . . . . . . . . 10 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹 ∧ 𝑡 ∈ 𝐴))
9	6, 1	opelcnv 4517	. . . . . . . . . . 11 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ↔ ⟨𝑡, 𝑠⟩ ∈ 𝐹)
10	9	anbi1i 431	. . . . . . . . . 10 ⊢ ((⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (⟨𝑡, 𝑠⟩ ∈ 𝐹 ∧ 𝑡 ∈ 𝐴))
11	8, 10	bitr4i 176	. . . . . . . . 9 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
12	7, 11	bitri 173	. . . . . . . 8 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
13	12	anbi2i 430	. . . . . . 7 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑠 ∈ 𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)))
14		anass 381	. . . . . . 7 ⊢ (((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (𝑠 ∈ 𝐵 ∧ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)))
15	13, 14	bitr4i 176	. . . . . 6 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
16	15	exbii 1496	. . . . 5 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
17		19.41v 1782	. . . . 5 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
18	16, 17	bitri 173	. . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
19	4, 5, 18	3bitr4ri 202	. . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
20	2, 19	bitri 173	. 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴))
21	20	eqriv 2037	1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)

Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∩ cin 2916  ⟨cop 3378  ^◡ccnv 4344   ↾ cres 4347   “ cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)