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 5009 | . 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 5011 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵 = (𝐶 ∪ 𝐷)) |
5 | 4 | uneq2d 4132 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ {𝑋, 𝑌}𝐵) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷))) |
6 | 1, 5 | syl5eq 2867 | 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 398 = wceq 1536 ∈ wcel 2113 ∪ cun 3927 {cpr 4562 ∪ 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-3an 1084 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-ral 3142 df-rex 3143 df-v 3493 df-sbc 3769 df-un 3934 df-in 3936 df-ss 3945 df-sn 4561 df-pr 4563 df-iun 4914 |
This theorem is referenced by: (None) |
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