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Mirrors > Home > MPE Home > Th. List > fnsng | Structured version Visualization version GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
fnsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 6398 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
2 | dmsnopg 6063 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
3 | 2 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom {〈𝐴, 𝐵〉} = {𝐴}) |
4 | df-fn 6351 | . 2 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} ↔ (Fun {〈𝐴, 𝐵〉} ∧ dom {〈𝐴, 𝐵〉} = {𝐴})) | |
5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-fun 6350 df-fn 6351 |
This theorem is referenced by: fnsn 6405 fnunsn 6457 fvsnun2 6937 fsnunfv 6941 mat1dimscm 21012 m1detdiag 21134 actfunsnf1o 31774 actfunsnrndisj 31775 breprexplema 31800 noextenddif 33072 noextendlt 33073 noextendgt 33074 fnsnbt 38998 |
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