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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 6082 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
3 | velsn 4644 | . . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵)) | |
4 | eqcom 2739 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
5 | 3, 4 | bitri 274 | . . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦) |
6 | fnbrfvb 6944 | . . . 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 2868 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)}) |
9 | 2, 8 | eqtr2d 2773 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 {csn 4628 class class class wbr 5148 “ cima 5679 Fn wfn 6538 ‘cfv 6543 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: fnimapr 6975 funfv 6978 fvco2 6988 fvimacnvi 7053 fvimacnvALT 7058 fsn2 7133 fparlem3 8099 fparlem4 8100 suppval1 8151 suppsnop 8162 domunsncan 9071 imafi 9174 phplem2 9207 phplem4OLD 9219 domunfican 9319 fiint 9323 infdifsn 9651 cantnfp1lem3 9674 resunimafz0 14403 symgfixelsi 19302 dprdf1o 19901 frlmlbs 21351 f1lindf 21376 cnt1 22853 xkohaus 23156 xkoptsub 23157 ustuqtop3 23747 bday1s 27329 old1 27367 madeoldsuc 27376 fnimatp 31897 eulerpartlemmf 33369 poimirlem4 36487 poimirlem6 36489 poimirlem7 36490 poimirlem9 36492 poimirlem13 36496 poimirlem14 36497 poimirlem16 36499 poimirlem19 36502 grpokerinj 36756 fnimasnd 41054 k0004lem3 42890 funcoressn 45742 |
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