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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 515 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
4 | f1sng 6631 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
5 | f1f 6549 | . 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 399 ∈ wcel 2111 {csn 4525 〈cop 4531 ⟶wf 6320 –1-1→wf1 6321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 |
This theorem is referenced by: 1fv 13021 snopiswrd 13866 frgpcyg 20265 mat1dimmul 21081 pt1hmeo 22411 upgr1e 26906 1hevtxdg1 27296 wlkp1 27471 wlkl0 28152 reprsuc 31996 breprexplema 32011 satfv1lem 32722 frlmsnic 39453 nnsum3primesprm 44308 0aryfvalel 45048 fv1arycl 45051 1arympt1fv 45053 1arymaptfo 45057 |
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