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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 553 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
4 | f1sng 6090 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
5 | f1f 6014 | . 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 383 ∈ wcel 1977 {csn 4125 〈cop 4131 ⟶wf 5800 –1-1→wf1 5801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: 1fv 12327 snopiswrd 13169 frgpcyg 19741 mat1dimmul 20101 pt1hmeo 21419 upgr1e 25779 nnsum3primesprm 40206 1hevtxdg1 40721 1wlkp1 40890 |
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