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Mirrors > Home > MPE Home > Th. List > Mathboxes > rtrclexlem | Structured version Visualization version GIF version |
Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.) |
Ref | Expression |
---|---|
rtrclexlem | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7907 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7908 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | 1, 2 | unexd 7754 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V) |
4 | sqxpexg 7755 | . . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∈ V → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ 𝑉 → ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) |
6 | unexg 7749 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅)) ∈ V) → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V) | |
7 | 5, 6 | mpdan 685 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3463 ∪ cun 3937 × cxp 5670 dom cdm 5672 ran crn 5673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 |
This theorem is referenced by: rtrclex 43112 |
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