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Mirrors > Home > NFE Home > Th. List > fconstopab | GIF version |
Description: Representation of a constant function using ordered pairs. (Contributed by NM, 12-Oct-1999.) |
Ref | Expression |
---|---|
fconstopab | ⊢ (A × {B}) = {〈x, y〉 ∣ (x ∈ A ∧ y = B)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4784 | . 2 ⊢ (A × {B}) = {〈x, y〉 ∣ (x ∈ A ∧ y ∈ {B})} | |
2 | df-sn 3741 | . . . . 5 ⊢ {B} = {y ∣ y = B} | |
3 | 2 | abeq2i 2460 | . . . 4 ⊢ (y ∈ {B} ↔ y = B) |
4 | 3 | anbi2i 675 | . . 3 ⊢ ((x ∈ A ∧ y ∈ {B}) ↔ (x ∈ A ∧ y = B)) |
5 | 4 | opabbii 4626 | . 2 ⊢ {〈x, y〉 ∣ (x ∈ A ∧ y ∈ {B})} = {〈x, y〉 ∣ (x ∈ A ∧ y = B)} |
6 | 1, 5 | eqtri 2373 | 1 ⊢ (A × {B}) = {〈x, y〉 ∣ (x ∈ A ∧ y = B)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 {csn 3737 {copab 4622 × cxp 4770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-sn 3741 df-opab 4623 df-xp 4784 |
This theorem is referenced by: fconst 5250 fopabsn 5441 fconstmpt 5681 |
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