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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunsn | Structured version Visualization version GIF version |
Description: Indexed union of a singleton. Compare dfiun2 4958 and rnmpt 5827. (Contributed by Steven Nguyen, 7-Jun-2023.) |
Ref | Expression |
---|---|
iunsn | ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4921 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} | |
2 | velsn 4583 | . . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
3 | 2 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) |
4 | 3 | abbii 2886 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
5 | 1, 4 | eqtri 2844 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 {csn 4567 ∪ ciun 4919 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-sn 4568 df-iun 4921 |
This theorem is referenced by: dfqs3 39172 |
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