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| Mirrors > Home > MPE Home > Th. List > iunxprg | Structured version Visualization version GIF version | ||
| Description: A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
| Ref | Expression |
|---|---|
| iunxprg.1 | ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) |
| iunxprg.2 | ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) |
| Ref | Expression |
|---|---|
| iunxprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pr 4629 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 2 | iuneq1 5008 | . . . 4 ⊢ ({𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 |
| 4 | iunxun 5094 | . . 3 ⊢ ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) | |
| 5 | 3, 4 | eqtri 2765 | . 2 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) |
| 6 | iunxprg.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
| 7 | 6 | iunxsng 5090 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷) |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷) |
| 9 | iunxprg.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
| 10 | 9 | iunxsng 5090 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸) |
| 11 | 10 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸) |
| 12 | 8, 11 | uneq12d 4169 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) = (𝐷 ∪ 𝐸)) |
| 13 | 5, 12 | eqtrid 2789 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 {cpr 4628 ∪ 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-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-iun 4993 |
| This theorem is referenced by: iunxunpr 32580 ovnsubadd2lem 46660 rnfdmpr 47293 |
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