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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 5828 | . 2 ⊢ ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
2 | df-mpt 5149 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
3 | 2 | rneqi 5809 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
4 | df-rex 3146 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})) | |
5 | 4 | abbii 2888 | . 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
6 | 1, 3, 5 | 3eqtr4i 2856 | 1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ∃wrex 3141 {csn 4569 {copab 5130 ↦ cmpt 5148 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: f1omptsnlem 34619 mptsnunlem 34621 dissneqlem 34623 |
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