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Mirrors > Home > MPE Home > Th. List > trclexlem | Structured version Visualization version GIF version |
Description: Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.) |
Ref | Expression |
---|---|
trclexlem | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7844 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7845 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | 1, 2 | xpexd 7689 | . 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V) |
4 | unexg 7687 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) | |
5 | 3, 4 | mpdan 686 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3447 ∪ cun 3912 × cxp 5635 dom cdm 5637 ran crn 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: trclublem 14889 trclfv 14894 cnvtrcl0 41990 |
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