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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 1950 | . . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) | |
2 | vex 3444 | . . . . . . . 8 ⊢ 𝑠 ∈ V | |
3 | 2 | opelresi 5826 | . . . . . . 7 ⊢ (〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 〈𝑡, 𝑠〉 ∈ 𝐹)) |
4 | vex 3444 | . . . . . . . 8 ⊢ 𝑡 ∈ V | |
5 | 2, 4 | opelcnv 5716 | . . . . . . 7 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ 〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴)) |
6 | 2, 4 | opelcnv 5716 | . . . . . . . 8 ⊢ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ↔ 〈𝑡, 𝑠〉 ∈ 𝐹) |
7 | 6 | anbi2ci 627 | . . . . . . 7 ⊢ ((〈𝑠, 𝑡〉 ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 〈𝑡, 𝑠〉 ∈ 𝐹)) |
8 | 3, 5, 7 | 3bitr4i 306 | . . . . . 6 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ∧ 𝑡 ∈ 𝐴)) |
9 | 8 | bianass 641 | . . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
10 | 9 | exbii 1849 | . . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
11 | 4 | elima3 5903 | . . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹)) |
12 | 11 | anbi1i 626 | . . . 4 ⊢ ((𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
13 | 1, 10, 12 | 3bitr4i 306 | . . 3 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) |
14 | 4 | elima3 5903 | . . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴))) |
15 | elin 3897 | . . 3 ⊢ (𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴) ↔ (𝑡 ∈ (◡𝐹 “ 𝐵) ∧ 𝑡 ∈ 𝐴)) | |
16 | 13, 14, 15 | 3bitr4i 306 | . 2 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ 𝑡 ∈ ((◡𝐹 “ 𝐵) ∩ 𝐴)) |
17 | 16 | eqriv 2795 | 1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∩ cin 3880 〈cop 4531 ◡ccnv 5518 ↾ cres 5521 “ cima 5522 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: fimacnvinrn 6817 ramub2 16340 ramub1lem2 16353 cnrest 21890 kgencn 22161 kgencn3 22163 xkoptsub 22259 qtopres 22303 qtoprest 22322 mbfid 24239 mbfres 24248 1stpreima 30466 2ndpreima 30467 gsumhashmul 30741 cvmsss2 32634 lmhmlnmsplit 40031 |
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