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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq1i | Structured version Visualization version GIF version |
Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
iuneq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
iuneq1i | ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1i.1 | . . . . . 6 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2831 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | anbi1i 624 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)) |
4 | 3 | rexbii2 3088 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
5 | 4 | abbii 2807 | . 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} |
6 | df-iun 4998 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
7 | df-iun 4998 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} | |
8 | 5, 6, 7 | 3eqtr4i 2773 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 ∪ ciun 4996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-iun 4998 |
This theorem is referenced by: ovolval4lem1 46605 |
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