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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 1953 | . . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) | |
2 | vex 3436 | . . . . . . . 8 ⊢ 𝑠 ∈ V | |
3 | 2 | opelresi 5899 | . . . . . . 7 ⊢ (〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 〈𝑡, 𝑠〉 ∈ 𝐹)) |
4 | vex 3436 | . . . . . . . 8 ⊢ 𝑡 ∈ V | |
5 | 2, 4 | opelcnv 5790 | . . . . . . 7 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ 〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴)) |
6 | 2, 4 | opelcnv 5790 | . . . . . . . 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 303 | . . . . . 6 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)) |
9 | 8 | bianass 639 | . . . . 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 5976 | . . . . 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 303 | . . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) |
14 | 4 | elima3 5976 | . . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴))) |
15 | elin 3903 | . . 3 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) | |
16 | 13, 14, 15 | 3bitr4i 303 | . 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴)) |
17 | 16 | eqriv 2735 | 1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∩ cin 3886 〈cop 4567 ◡ccnv 5588 ↾ cres 5591 “ cima 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: fimacnvinrn 6949 ramub2 16715 ramub1lem2 16728 cnrest 22436 kgencn 22707 kgencn3 22709 xkoptsub 22805 qtopres 22849 qtoprest 22868 mbfid 24799 mbfres 24808 1stpreima 31039 2ndpreima 31040 gsumhashmul 31316 cvmsss2 33236 lmhmlnmsplit 40912 |
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