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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 2833 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | 
| 3 | 2 | anbi1i 624 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶)) | 
| 4 | 3 | rexbii2 3090 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) | 
| 5 | 4 | abbii 2809 | . 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} | 
| 6 | df-iun 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} | |
| 7 | df-iun 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶} | |
| 8 | 5, 6, 7 | 3eqtr4i 2775 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ∪ ciun 4991 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-iun 4993 | 
| This theorem is referenced by: ovolval4lem1 46664 | 
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