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Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq12f | Structured version Visualization version GIF version |
Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
iuneq12f.1 | ⊢ Ⅎ𝑥𝐴 |
iuneq12f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
iuneq12f | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2 4929 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐴 𝐷) | |
2 | iuneq12f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | iuneq12f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | iuneq2f 35315 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷) |
5 | 1, 4 | sylan9eqr 2875 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnfc 2958 ∀wral 3135 ∪ ciun 4910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-in 3940 df-ss 3949 df-iun 4912 |
This theorem is referenced by: (None) |
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