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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptsn | Structured version Visualization version GIF version |
Description: The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.) |
Ref | Expression |
---|---|
rnmptsn | ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnopab 5791 | . 2 ⊢ ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
2 | df-mpt 5108 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
3 | 2 | rneqi 5774 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
4 | df-rex 3059 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})) | |
5 | 4 | abbii 2803 | . 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
6 | 1, 3, 5 | 3eqtr4i 2771 | 1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2113 {cab 2716 ∃wrex 3054 {csn 4513 {copab 5089 ↦ cmpt 5107 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rex 3059 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-mpt 5108 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: f1omptsnlem 35119 mptsnunlem 35121 dissneqlem 35123 |
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