Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > resiun1 | Structured version Visualization version GIF version |
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.) |
Ref | Expression |
---|---|
resiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin1 4994 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
2 | df-res 5567 | . . . 4 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))) |
4 | 3 | iuneq2i 4940 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) |
5 | df-res 5567 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
6 | 1, 4, 5 | 3eqtr4ri 2855 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ∪ ciun 4919 × cxp 5553 ↾ cres 5557 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-iun 4921 df-res 5567 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |