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| Mirrors > Home > MPE Home > Th. List > fsnd | Structured version Visualization version GIF version | ||
| Description: A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.) |
| Ref | Expression |
|---|---|
| fsnd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| fsnd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| Ref | Expression |
|---|---|
| fsnd | ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fsnd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | fsnd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
| 4 | f1sng 6890 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
| 5 | f1f 6804 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) | |
| 6 | 3, 4, 5 | 3syl 18 | 1 ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {csn 4626 〈cop 4632 ⟶wf 6557 –1-1→wf1 6558 |
| 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-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: 1fv 13687 snopiswrd 14561 frgpcyg 21592 mat1dimmul 22482 pt1hmeo 23814 upgr1e 29130 1hevtxdg1 29524 wlkp1 29699 wlkl0 30386 reprsuc 34630 breprexplema 34645 satfv1lem 35367 frlmsnic 42550 fsetsniunop 47061 nnsum3primesprm 47777 0aryfvalel 48555 fv1arycl 48558 1arympt1fv 48560 1arymaptfo 48564 |
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