Theorem cnvresima 6229

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 1953	. . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
2		vex 3478	. . . . . . . 8 ⊢ 𝑠 ∈ V
3	2	opelresi 5989	. . . . . . 7 ⊢ (⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ ⟨𝑡, 𝑠⟩ ∈ 𝐹))
4		vex 3478	. . . . . . . 8 ⊢ 𝑡 ∈ V
5	2, 4	opelcnv 5881	. . . . . . 7 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ ⟨𝑡, 𝑠⟩ ∈ (𝐹 ↾ 𝐴))
6	2, 4	opelcnv 5881	. . . . . . . 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 302	. . . . . 6 ⊢ (⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴) ↔ (⟨𝑠, 𝑡⟩ ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴))
9	8	bianass 640	. . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
10	9	exbii 1850	. . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴))
11	4	elima3 6066	. . . . 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 302	. . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴))
14	4	elima3 6066	. . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ ⟨𝑠, 𝑡⟩ ∈ ◡(𝐹 ↾ 𝐴)))
15		elin 3964	. . 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 2729	1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)

Syntax hints:   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ∩ cin 3947  ⟨cop 4634  ^◡ccnv 5675   ↾ cres 5678   “ cima 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by:  fimacnvinrn  7073  ramub2  16946  ramub1lem2  16959  cnrest  22788  kgencn  23059  kgencn3  23061  xkoptsub  23157  qtopres  23201  qtoprest  23220  mbfid  25151  mbfres  25160  1stpreima  31923  2ndpreima  31924  gsumhashmul  32203  cvmsss2  34260  lmhmlnmsplit  41819