Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ressn | GIF version |
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
ressn | ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4917 | . 2 ⊢ Rel (𝐴 ↾ {𝐵}) | |
2 | relxp 4718 | . 2 ⊢ Rel ({𝐵} × (𝐴 “ {𝐵})) | |
3 | ancom 264 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
4 | vex 2733 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 2733 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | elimasn 4976 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | elsni 3599 | . . . . . . . . 9 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
8 | 7 | sneqd 3594 | . . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → {𝑥} = {𝐵}) |
9 | 8 | imaeq2d 4951 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} → (𝐴 “ {𝑥}) = (𝐴 “ {𝐵})) |
10 | 9 | eleq2d 2240 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → (𝑦 ∈ (𝐴 “ {𝑥}) ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
11 | 6, 10 | bitr3id 193 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝑦 ∈ (𝐴 “ {𝐵}))) |
12 | 11 | pm5.32i 451 | . . . 4 ⊢ ((𝑥 ∈ {𝐵} ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
13 | 3, 12 | bitri 183 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵}) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) |
14 | 5 | opelres 4894 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ {𝐵})) |
15 | opelxp 4639 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵})) ↔ (𝑥 ∈ {𝐵} ∧ 𝑦 ∈ (𝐴 “ {𝐵}))) | |
16 | 13, 14, 15 | 3bitr4i 211 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 ↾ {𝐵}) ↔ 〈𝑥, 𝑦〉 ∈ ({𝐵} × (𝐴 “ {𝐵}))) |
17 | 1, 2, 16 | eqrelriiv 4703 | 1 ⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3581 〈cop 3584 × cxp 4607 ↾ cres 4611 “ cima 4612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |