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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 5094 | . 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 5096 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷)) | 
| 5 | 4 | uneq2d 4168 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) | 
| 6 | 1, 5 | eqtrid 2789 | 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 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {cpr 4628 ∪ ciun 4991 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-iun 4993 | 
| This theorem is referenced by: (None) | 
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