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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 5964 | . 2 ⊢ ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
| 2 | df-mpt 5225 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
| 3 | 2 | rneqi 5947 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | 
| 4 | df-rex 3070 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})) | |
| 5 | 4 | abbii 2808 | . 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | 
| 6 | 1, 3, 5 | 3eqtr4i 2774 | 1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {csn 4625 {copab 5204 ↦ cmpt 5224 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: f1omptsnlem 37338 mptsnunlem 37340 dissneqlem 37342 | 
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