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Mirrors > Home > NFE Home > Th. List > mptv | GIF version |
Description: Function with universal domain in maps-to notation. (Contributed by set.mm contributors, 16-Aug-2013.) |
Ref | Expression |
---|---|
mptv | ⊢ (x ∈ V ↦ B) = {〈x, y〉 ∣ y = B} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 5652 | . 2 ⊢ (x ∈ V ↦ B) = {〈x, y〉 ∣ (x ∈ V ∧ y = B)} | |
2 | vex 2862 | . . . 4 ⊢ x ∈ V | |
3 | 2 | biantrur 492 | . . 3 ⊢ (y = B ↔ (x ∈ V ∧ y = B)) |
4 | 3 | opabbii 4626 | . 2 ⊢ {〈x, y〉 ∣ y = B} = {〈x, y〉 ∣ (x ∈ V ∧ y = B)} |
5 | 1, 4 | eqtr4i 2376 | 1 ⊢ (x ∈ V ↦ B) = {〈x, y〉 ∣ y = B} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {copab 4622 ↦ cmpt 5651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-v 2861 df-opab 4623 df-mpt 5652 |
This theorem is referenced by: csucex 6259 brcsuc 6260 frecsuc 6322 |
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