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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 | df-mpt 4730 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
2 | 1 | rneqi 5352 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
3 | rnopab 5370 | . . 3 ⊢ ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
4 | 2, 3 | eqtri 2644 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
5 | df-rex 2918 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})) | |
6 | 5 | abbii 2739 | . 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
7 | 4, 6 | eqtr4i 2647 | 1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cab 2608 ∃wrex 2913 {csn 4177 {copab 4712 ↦ cmpt 4729 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: f1omptsnlem 33183 mptsnunlem 33185 dissneqlem 33187 |
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