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| 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 6101 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | 
| 3 | velsn 4641 | . . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵)) | |
| 4 | eqcom 2743 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
| 5 | 3, 4 | bitri 275 | . . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦) | 
| 6 | fnbrfvb 6958 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦)) | |
| 7 | 5, 6 | bitr2id 284 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝑦 ↔ 𝑦 ∈ {(𝐹‘𝐵)})) | 
| 8 | 7 | eqabcdv 2875 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)}) | 
| 9 | 2, 8 | eqtr2d 2777 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 {csn 4625 class class class wbr 5142 “ cima 5687 Fn wfn 6555 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: fnimapr 6991 fnimatpd 6992 funfv 6995 fvco2 7005 fvimacnvi 7071 fvimacnvALT 7076 fsn2 7155 fparlem3 8140 fparlem4 8141 suppval1 8192 suppsnop 8204 domunsncan 9113 phplem2 9246 phplem4OLD 9258 imafiOLD 9355 domunfican 9362 fiint 9367 fiintOLD 9368 infdifsn 9698 cantnfp1lem3 9721 resunimafz0 14485 symgfixelsi 19454 dprdf1o 20053 frlmlbs 21818 f1lindf 21843 cnt1 23359 xkohaus 23662 xkoptsub 23663 ustuqtop3 24253 bday1s 27877 old1 27915 madeoldsuc 27924 n0sbday 28355 zscut 28394 pw2bday 28419 zs12bday 28425 cyclnumvtx 29821 eulerpartlemmf 34378 poimirlem4 37632 poimirlem6 37634 poimirlem7 37635 poimirlem9 37637 poimirlem13 37641 poimirlem14 37642 poimirlem16 37644 poimirlem19 37647 grpokerinj 37901 fnimasnd 42272 k0004lem3 44167 funcoressn 47059 cycl3grtri 47919 | 
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