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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 4572 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | iuneq1 4937 | . . . 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 5018 | . . 3 ⊢ ∪ 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) | |
5 | 3, 4 | eqtri 2846 | . 2 ⊢ ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) |
6 | iunxprg.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
7 | 6 | iunxsng 5014 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴}𝐶 = 𝐷) |
9 | iunxprg.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
10 | 9 | iunxsng 5014 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸) |
11 | 10 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐵}𝐶 = 𝐸) |
12 | 8, 11 | uneq12d 4142 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∪ 𝑥 ∈ {𝐴}𝐶 ∪ ∪ 𝑥 ∈ {𝐵}𝐶) = (𝐷 ∪ 𝐸)) |
13 | 5, 12 | syl5eq 2870 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 {csn 4569 {cpr 4571 ∪ ciun 4921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-sbc 3775 df-un 3943 df-in 3945 df-ss 3954 df-sn 4570 df-pr 4572 df-iun 4923 |
This theorem is referenced by: iunxunpr 30321 ovnsubadd2lem 42934 rnfdmpr 43487 |
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