| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > funsn | Structured version Visualization version GIF version | ||
| Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
| Ref | Expression |
|---|---|
| funsn.1 | ⊢ 𝐴 ∈ V |
| funsn.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| funsn | ⊢ Fun {〈𝐴, 𝐵〉} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funsn.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | funsn.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | funsng 6617 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {〈𝐴, 𝐵〉}) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ Fun {〈𝐴, 𝐵〉} |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: funtp 6623 fun0 6631 funop 7169 funsndifnop 7171 wfrlem13OLD 8361 dcomex 10487 axdc3lem4 10493 cnfldfunALT 21379 cnfldfunALTOLD 21392 cnfldfunALTOLDOLD 21393 bnj1421 35056 funop1 47295 |
| Copyright terms: Public domain | W3C validator |