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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 6619 | . 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 2106 Vcvv 3478 {csn 4631 〈cop 4637 Fun wfun 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-fun 6565 |
This theorem is referenced by: funtp 6625 fun0 6633 funop 7169 funsndifnop 7171 wfrlem13OLD 8360 dcomex 10485 axdc3lem4 10491 cnfldfunALT 21397 cnfldfunALTOLD 21410 cnfldfunALTOLDOLD 21411 bnj1421 35035 funop1 47233 |
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