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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 7914 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7915 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | 1, 2 | xpexd 7759 | . 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V) |
4 | unexg 7757 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) | |
5 | 3, 4 | mpdan 685 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3462 ∪ cun 3945 × cxp 5680 dom cdm 5682 ran crn 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: trclublem 15000 trclfv 15005 cnvtrcl0 43293 |
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