Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunxunsn | Structured version Visualization version GIF version |
Description: Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
iunxunsn.1 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iunxunsn | ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxun 5016 | . 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) | |
2 | iunxunsn.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) | |
3 | 2 | iunxsng 5012 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ {𝑋}𝐵 = 𝐶) |
4 | 3 | uneq2d 4139 | . 2 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)) |
5 | 1, 4 | syl5eq 2868 | 1 ⊢ (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 {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-3an 1085 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-ral 3143 df-rex 3144 df-v 3496 df-sbc 3773 df-un 3941 df-sn 4568 df-iun 4921 |
This theorem is referenced by: (None) |
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