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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 6408 | . 2 ⊢ Fun ◡{〈𝐵, 𝐴〉} | |
2 | cnvsng 6066 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) | |
3 | 2 | ancoms 462 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{〈𝐵, 𝐴〉} = {〈𝐴, 𝐵〉}) |
4 | 3 | funeqd 6380 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Fun ◡{〈𝐵, 𝐴〉} ↔ Fun {〈𝐴, 𝐵〉})) |
5 | 1, 4 | mpbii 236 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {csn 4527 〈cop 4533 ◡ccnv 5535 Fun wfun 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-fun 6360 |
This theorem is referenced by: fnsng 6410 funsn 6411 funprg 6412 funtpg 6413 fvsng 6973 tfrlem10 8101 snopfsupp 8986 funsnfsupp 8987 setsfun 16700 setsfun0 16701 strle1 16776 p1evtxdeqlem 27554 trlsegvdeglem3 28259 bnj519 32381 bnj150 32523 noextend 33555 |
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