Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 5002 | . 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 5004 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷)) |
5 | 4 | uneq2d 4077 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) |
6 | 1, 5 | syl5eq 2790 | 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 399 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 {cpr 4543 ∪ ciun 4904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-v 3410 df-un 3871 df-in 3873 df-ss 3883 df-sn 4542 df-pr 4544 df-iun 4906 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |