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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbviunvw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbviunvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
cbviunvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbviunvw2 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviunvw2.2 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | cbviunvw2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
3 | 2 | eleq2d 2823 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)) |
4 | 1, 3 | cbvrexvw2 36170 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷) |
5 | 4 | abbii 2805 | . 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷} |
6 | df-iun 5000 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
7 | df-iun 5000 | . 2 ⊢ ∪ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷} | |
8 | 5, 6, 7 | 3eqtr4i 2771 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 {cab 2710 ∃wrex 3066 ∪ ciun 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rex 3067 df-iun 5000 |
This theorem is referenced by: (None) |
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