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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 6618 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 Fn wfn 6556 |
| 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-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: f1osn 6888 fnsnb 7186 frrlem11 8321 frrlem12 8322 elixpsn 8977 axdc3lem4 10493 hashf1lem1 14494 axlowdimlem8 28964 axlowdimlem9 28965 axlowdimlem11 28967 axlowdimlem12 28968 bnj927 34783 cvmliftlem4 35293 cvmliftlem5 35294 finixpnum 37612 poimirlem3 37630 |
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