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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 6267 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ Fun {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2079 Vcvv 3432 {csn 4466 〈cop 4472 Fun wfun 6211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-opab 5019 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-fun 6219 |
This theorem is referenced by: funtp 6273 fun0 6281 funop 6765 funsndifnop 6767 fvsnOLD 6798 wfrlem13 7810 dcomex 9704 axdc3lem4 9710 cnfldfun 20227 bnj1421 31884 funop1 42952 |
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