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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 4951 and rnmpt 5820. (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 4914 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} | |
2 | velsn 4576 | . . . 4 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
3 | 2 | rexbii 3246 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵) |
4 | 3 | abbii 2885 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {𝐵}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
5 | 1, 4 | eqtri 2843 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 {cab 2798 ∃wrex 3138 {csn 4560 ∪ ciun 4912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 df-v 3493 df-sn 4561 df-iun 4914 |
This theorem is referenced by: dfqs3 39200 |
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