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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 5951 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
2 | 1 | adantl 485 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
3 | velsn 4557 | . . . . 5 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ 𝑦 = (𝐹‘𝐵)) | |
4 | eqcom 2744 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
5 | 3, 4 | bitri 278 | . . . 4 ⊢ (𝑦 ∈ {(𝐹‘𝐵)} ↔ (𝐹‘𝐵) = 𝑦) |
6 | fnbrfvb 6765 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦)) | |
7 | 5, 6 | bitr2id 287 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝑦 ↔ 𝑦 ∈ {(𝐹‘𝐵)})) |
8 | 7 | abbi1dv 2875 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝐵𝐹𝑦} = {(𝐹‘𝐵)}) |
9 | 2, 8 | eqtr2d 2778 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {cab 2714 {csn 4541 class class class wbr 5053 “ cima 5554 Fn wfn 6375 ‘cfv 6380 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-fv 6388 |
This theorem is referenced by: fnimapr 6795 funfv 6798 fvco2 6808 fvimacnvi 6872 fvimacnvALT 6877 fsn2 6951 fparlem3 7882 fparlem4 7883 suppval1 7909 suppsnop 7920 domunsncan 8745 phplem4 8828 imafi 8853 domunfican 8944 fiint 8948 infdifsn 9272 cantnfp1lem3 9295 resunimafz0 14009 symgfixelsi 18827 dprdf1o 19419 frlmlbs 20759 f1lindf 20784 cnt1 22247 xkohaus 22550 xkoptsub 22551 ustuqtop3 23141 fnimatp 30734 eulerpartlemmf 32054 bday1s 33762 madeoldsuc 33804 poimirlem4 35518 poimirlem6 35520 poimirlem7 35521 poimirlem9 35523 poimirlem13 35527 poimirlem14 35528 poimirlem16 35530 poimirlem19 35533 grpokerinj 35788 fnimasnd 39922 k0004lem3 41436 funcoressn 44208 |
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