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Mirrors > Home > MPE Home > Th. List > resiun1OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of resiun1 5336 as of 25-Aug-2021. (Contributed by Mario Carneiro, 29-May-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
resiun1OLD | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin2 4520 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | df-res 5050 | . . . . 5 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
3 | incom 3767 | . . . . 5 ⊢ (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵) | |
4 | 2, 3 | eqtri 2632 | . . . 4 ⊢ (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵)) |
6 | 5 | iuneq2i 4475 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) |
7 | df-res 5050 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
8 | incom 3767 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
9 | 7, 8 | eqtri 2632 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) |
10 | 1, 6, 9 | 3eqtr4ri 2643 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ∪ ciun 4455 × cxp 5036 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iun 4457 df-res 5050 |
This theorem is referenced by: (None) |
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