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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iuneq12daf | Structured version Visualization version GIF version | ||
| Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.) |
| Ref | Expression |
|---|---|
| iuneq12daf.1 | ⊢ Ⅎ𝑥𝜑 |
| iuneq12daf.2 | ⊢ Ⅎ𝑥𝐴 |
| iuneq12daf.3 | ⊢ Ⅎ𝑥𝐵 |
| iuneq12daf.4 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| iuneq12daf.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| iuneq12daf | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iuneq12daf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 2 | iuneq12daf.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) | |
| 3 | 2 | eleq2d 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
| 4 | 1, 3 | rexbida 3272 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷)) |
| 5 | iuneq12daf.4 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 6 | iuneq12daf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 7 | iuneq12daf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
| 8 | 6, 7 | rexeqf 3354 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
| 9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
| 10 | 4, 9 | bitrd 279 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
| 11 | 10 | alrimiv 1927 | . 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
| 12 | abbi 2807 | . . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}) | |
| 13 | df-iun 4993 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
| 14 | df-iun 4993 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷} | |
| 15 | 12, 13, 14 | 3eqtr4g 2802 | . 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
| 16 | 11, 15 | syl 17 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∃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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-iun 4993 |
| This theorem is referenced by: measvunilem0 34214 |
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