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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunxunpr | Structured version Visualization version GIF version |
Description: Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
iunxunsn.1 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) |
iunxunpr.2 | ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
iunxunpr | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxun 5091 | . 2 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) | |
2 | iunxunsn.1 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶) | |
3 | iunxunpr.2 | . . . 4 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) | |
4 | 2, 3 | iunxprg 5093 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷)) |
5 | 4 | uneq2d 4160 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) |
6 | 1, 5 | eqtrid 2784 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∪ cun 3943 {cpr 4625 ∪ ciun 4991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-v 3476 df-un 3950 df-in 3952 df-ss 3962 df-sn 4624 df-pr 4626 df-iun 4993 |
This theorem is referenced by: (None) |
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