Mathbox for Thierry Arnoux |
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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 2898 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
4 | 1, 3 | rexbida 3318 | . . . 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 3399 | . . . . 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 281 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
11 | 10 | alrimiv 1924 | . 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
12 | abbi1 2884 | . . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}) | |
13 | df-iun 4914 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
14 | df-iun 4914 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷} | |
15 | 12, 13, 14 | 3eqtr4g 2881 | . 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 208 ∧ wa 398 ∀wal 1531 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∃wrex 3139 ∪ ciun 4912 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-iun 4914 |
This theorem is referenced by: measvunilem0 31467 |
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