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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 514 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
4 | f1sng 6658 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
5 | f1f 6577 | . 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 398 ∈ wcel 2114 {csn 4569 〈cop 4575 ⟶wf 6353 –1-1→wf1 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: 1fv 13029 snopiswrd 13873 frgpcyg 20722 mat1dimmul 21087 pt1hmeo 22416 upgr1e 26900 1hevtxdg1 27290 wlkp1 27465 wlkl0 28148 reprsuc 31888 breprexplema 31903 satfv1lem 32611 frlmsnic 39156 nnsum3primesprm 43962 |
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