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 2819 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
4 | 1, 3 | rexbida 3230 | . . . 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 3303 | . . . . 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 282 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
11 | 10 | alrimiv 1935 | . 2 ⊢ (𝜑 → ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
12 | abbi1 2802 | . . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}) | |
13 | df-iun 4896 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
14 | df-iun 4896 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐷} | |
15 | 12, 13, 14 | 3eqtr4g 2799 | . 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 209 ∧ wa 399 ∀wal 1541 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 {cab 2712 Ⅎwnfc 2880 ∃wrex 3055 ∪ ciun 4894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-rex 3060 df-iun 4896 |
This theorem is referenced by: measvunilem0 31865 |
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