![]() |
Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq2f | Structured version Visualization version GIF version |
Description: Equality deduction for indexed union. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Ref | Expression |
---|---|
iuneq2f.1 | ⊢ Ⅎ𝑥𝐴 |
iuneq2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
iuneq2f | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | iuneq2f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2805 | . 2 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
5 | eqidd 2652 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
6 | 3, 1, 2, 4, 5 | iuneq12df 4576 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 Ⅎwnfc 2780 ∪ ciun 4552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rex 2947 df-iun 4554 |
This theorem is referenced by: iuneq12f 34102 |
Copyright terms: Public domain | W3C validator |