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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnrnafv | Structured version Visualization version GIF version |
Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6968. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
fnrnafv | ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafn5a 47110 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | |
2 | 1 | rneqd 5952 | . 2 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) |
3 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) | |
4 | 3 | rnmpt 5971 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)} |
5 | 2, 4 | 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 '''cafv 47067 |
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-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 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-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-aiota 47035 df-dfat 47069 df-afv 47070 |
This theorem is referenced by: afvelrnb 47113 afvelrnb0 47114 |
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