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Mirrors > Home > MPE Home > Th. List > fnrnfv | Structured version Visualization version GIF version |
Description: The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fnrnfv | ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5 6967 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
2 | rneq 5950 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
3 | 1, 2 | sylbi 217 | . 2 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
4 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) | |
5 | 4 | rnmpt 5971 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
6 | 3, 5 | eqtrdi 2791 | 1 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {cab 2712 ∃wrex 3068 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: fvelrnb 6969 fniinfv 6987 dffo3 7122 dffo3f 7126 fniunfv 7267 fnrnov 7606 pwcfsdom 10621 hauscmplem 23430 madef 27910 grpoinvf 30561 fpwrelmapffslem 32750 cshwrnid 32931 meascnbl 34200 omssubadd 34282 fvineqsneu 37394 fvineqsneq 37395 tfsconcatrn 43332 rnfdmpr 47231 fargshiftfo 47367 |
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