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Mirrors > Home > MPE Home > Th. List > fnsn | Structured version Visualization version GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
fnsn.1 | ⊢ 𝐴 ∈ V |
fnsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fnsn | ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | fnsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | fnsng 6406 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3494 {csn 4567 〈cop 4573 Fn wfn 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-fun 6357 df-fn 6358 |
This theorem is referenced by: f1osn 6654 fnsnb 6928 elixpsn 8501 axdc3lem4 9875 hashf1lem1 13814 axlowdimlem8 26735 axlowdimlem9 26736 axlowdimlem11 26738 axlowdimlem12 26739 bnj927 32040 cvmliftlem4 32535 cvmliftlem5 32536 frrlem11 33133 frrlem12 33134 finixpnum 34892 poimirlem3 34910 |
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