![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1954 | . . . 4 ⊢ (∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴) ↔ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) | |
2 | vex 3479 | . . . . . . . 8 ⊢ 𝑠 ∈ V | |
3 | 2 | opelresi 5984 | . . . . . . 7 ⊢ (〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 〈𝑡, 𝑠〉 ∈ 𝐹)) |
4 | vex 3479 | . . . . . . . 8 ⊢ 𝑡 ∈ V | |
5 | 2, 4 | opelcnv 5876 | . . . . . . 7 ⊢ (〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴) ↔ 〈𝑡, 𝑠〉 ∈ (𝐹 ↾ 𝐴)) |
6 | 2, 4 | opelcnv 5876 | . . . . . . . 8 ⊢ (〈𝑠, 𝑡〉 ∈ ◡𝐹 ↔ 〈𝑡, 𝑠〉 ∈ 𝐹) |
7 | 6 | anbi2ci 626 | . . . . . . 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 641 | . . . . 5 ⊢ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
10 | 9 | exbii 1851 | . . . 4 ⊢ (∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴)) ↔ ∃𝑠((𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹) ∧ 𝑡 ∈ 𝐴)) |
11 | 4 | elima3 6059 | . . . . 5 ⊢ (𝑡 ∈ (◡𝐹 “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡𝐹)) |
12 | 11 | anbi1i 625 | . . . 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 6059 | . . 3 ⊢ (𝑡 ∈ (◡(𝐹 ↾ 𝐴) “ 𝐵) ↔ ∃𝑠(𝑠 ∈ 𝐵 ∧ 〈𝑠, 𝑡〉 ∈ ◡(𝐹 ↾ 𝐴))) |
15 | elin 3962 | . . 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 2730 | 1 ⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∩ cin 3945 〈cop 4630 ◡ccnv 5671 ↾ cres 5674 “ cima 5675 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 |
This theorem is referenced by: fimacnvinrn 7061 ramub2 16934 ramub1lem2 16947 cnrest 22758 kgencn 23029 kgencn3 23031 xkoptsub 23127 qtopres 23171 qtoprest 23190 mbfid 25121 mbfres 25130 1stpreima 31899 2ndpreima 31900 gsumhashmul 32179 cvmsss2 34196 lmhmlnmsplit 41700 |
Copyright terms: Public domain | W3C validator |