![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iuneq12df | Structured version Visualization version GIF version |
Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.) |
Ref | Expression |
---|---|
iuneq12df.1 | ⊢ Ⅎ𝑥𝜑 |
iuneq12df.2 | ⊢ Ⅎ𝑥𝐴 |
iuneq12df.3 | ⊢ Ⅎ𝑥𝐵 |
iuneq12df.4 | ⊢ (𝜑 → 𝐴 = 𝐵) |
iuneq12df.5 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
iuneq12df | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq12df.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | iuneq12df.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | iuneq12df.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | iuneq12df.4 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | iuneq12df.5 | . . . . 5 ⊢ (𝜑 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2813 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
7 | 1, 2, 3, 4, 6 | rexeqbid 3347 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
8 | 7 | alrimiv 1922 | . 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
9 | abbi 2794 | . . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}) | |
10 | df-iun 4992 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
11 | df-iun 4992 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷} | |
12 | 9, 10, 11 | 3eqtr4g 2791 | . 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
13 | 8, 12 | syl 17 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 {cab 2703 Ⅎwnfc 2877 ∃wrex 3064 ∪ ciun 4990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-iun 4992 |
This theorem is referenced by: iunxdif3 5091 iundisjf 32329 aciunf1 32397 measvuni 33742 iuneq2f 37537 |
Copyright terms: Public domain | W3C validator |