![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnsnfv | Structured version Visualization version GIF version |
Description: Singleton of function value. (Contributed by NM, 22-May-1998.) (Proof shortened by Scott Fenton, 8-Aug-2024.) |
Ref | Expression |
---|---|
fnsnfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasng 6083 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
2 | 1 | adantl 480 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
3 | velsn 4645 | . . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵)) | |
4 | eqcom 2737 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
5 | 3, 4 | bitri 274 | . . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦) |
6 | fnbrfvb 6945 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦)) | |
7 | 5, 6 | bitr2id 283 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝑦 ↔ 𝑦 ∈ {(𝐹‘𝐵)})) |
8 | 7 | eqabcdv 2866 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)}) |
9 | 2, 8 | eqtr2d 2771 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {cab 2707 {csn 4629 class class class wbr 5149 “ cima 5680 Fn wfn 6539 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 |
This theorem is referenced by: fnimapr 6976 funfv 6979 fvco2 6989 fvimacnvi 7054 fvimacnvALT 7059 fsn2 7137 fparlem3 8104 fparlem4 8105 suppval1 8156 suppsnop 8167 domunsncan 9076 imafi 9179 phplem2 9212 phplem4OLD 9224 domunfican 9324 fiint 9328 infdifsn 9656 cantnfp1lem3 9679 resunimafz0 14410 symgfixelsi 19346 dprdf1o 19945 frlmlbs 21573 f1lindf 21598 cnt1 23076 xkohaus 23379 xkoptsub 23380 ustuqtop3 23970 bday1s 27567 old1 27605 madeoldsuc 27614 fnimatp 32167 eulerpartlemmf 33670 poimirlem4 36797 poimirlem6 36799 poimirlem7 36800 poimirlem9 36802 poimirlem13 36806 poimirlem14 36807 poimirlem16 36809 poimirlem19 36812 grpokerinj 37066 fnimasnd 41360 k0004lem3 43204 funcoressn 46052 |
Copyright terms: Public domain | W3C validator |