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Mirrors > Home > MPE Home > Th. List > funsng | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
Ref | Expression |
---|---|
funsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvsn 6406 | . 2 ⊢ Fun ◡{〈𝐵, 𝐴〉} | |
2 | cnvsng 6082 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) | |
3 | 2 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) |
4 | 3 | funeqd 6379 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ◡{〈𝐵, 𝐴〉} ↔ Fun {〈𝐴, 𝐵〉})) |
5 | 1, 4 | mpbii 235 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 〈cop 4575 ◡ccnv 5556 Fun wfun 6351 |
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-fun 6359 |
This theorem is referenced by: fnsng 6408 funsn 6409 funprg 6410 funtpg 6411 fvsng 6944 tfrlem10 8025 snopfsupp 8858 funsnfsupp 8859 setsfun 16520 setsfun0 16521 strle1 16594 p1evtxdeqlem 27296 trlsegvdeglem3 28003 bnj519 32008 bnj150 32150 noextend 33175 |
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