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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 4645 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
2 | 1 | rneqi 5273 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
3 | rnopab 5291 | . . 3 ⊢ ran {〈𝑥, 𝑢〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
5 | df-rex 2902 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})) | |
6 | 5 | abbii 2726 | . 2 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑢 = {𝑥})} |
7 | 4, 6 | eqtr4i 2635 | 1 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∃wrex 2897 {csn 4125 {copab 4642 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: f1omptsnlem 32359 mptsnunlem 32361 dissneqlem 32363 |
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