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Mirrors > Home > MPE Home > Th. List > cnvresima | Structured version Visualization version GIF version |
Description: An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.) |
Ref | Expression |
---|---|
cnvresima | ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.41v 1914 | . . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) | |
2 | vex 3203 | . . . . . . . 8 ⊢ 𝑠 ∈ V | |
3 | 2 | opelres 5401 | . . . . . . 7 ⊢ (〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴) ↔ (〈𝑡, 𝑠〉 ∈ 𝐹 ∧ 𝑡 ∈ 𝐴)) |
4 | vex 3203 | . . . . . . . 8 ⊢ 𝑡 ∈ V | |
5 | 2, 4 | opelcnv 5304 | . . . . . . 7 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ 〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴)) |
6 | 2, 4 | opelcnv 5304 | . . . . . . . 8 ⊢ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ↔ 〈𝑡, 𝑠〉 ∈ 𝐹) |
7 | 6 | anbi1i 731 | . . . . . . 7 ⊢ ((〈𝑠, 𝑡〉 ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (〈𝑡, 𝑠〉 ∈ 𝐹 ∧ 𝑡 ∈ 𝐴)) |
8 | 3, 5, 7 | 3bitr4i 292 | . . . . . 6 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)) |
9 | 8 | bianass 842 | . . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
10 | 9 | exbii 1774 | . . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
11 | 4 | elima3 5473 | . . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹)) |
12 | 11 | anbi1i 731 | . . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
13 | 1, 10, 12 | 3bitr4i 292 | . . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) |
14 | 4 | elima3 5473 | . . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴))) |
15 | elin 3796 | . . 3 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) | |
16 | 13, 14, 15 | 3bitr4i 292 | . 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴)) |
17 | 16 | eqriv 2619 | 1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∩ cin 3573 〈cop 4183 ◡ccnv 5113 ↾ cres 5116 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: fimacnvinrn 6348 ramub2 15718 ramub1lem2 15731 cnrest 21089 kgencn 21359 kgencn3 21361 xkoptsub 21457 qtopres 21501 qtoprest 21520 mbfid 23403 mbfres 23411 1stpreima 29484 2ndpreima 29485 cvmsss2 31256 lmhmlnmsplit 37657 |
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