Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5938 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 Fn wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: f1osn 6176 fnsnb 6432 fvsnun2 6449 elixpsn 7947 axdc3lem4 9275 hashf1lem1 13239 axlowdimlem8 25829 axlowdimlem9 25830 axlowdimlem11 25832 axlowdimlem12 25833 bnj927 30839 cvmliftlem4 31270 cvmliftlem5 31271 finixpnum 33394 poimirlem3 33412 |
Copyright terms: Public domain | W3C validator |