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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 7097 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7098 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | xpexg 6960 | . . 3 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 × ran 𝑅) ∈ V) | |
4 | 1, 2, 3 | syl2anc 693 | . 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V) |
5 | unexg 6959 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) | |
6 | 4, 5 | mpdan 702 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 × cxp 5112 dom cdm 5114 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: trclublem 13734 trclfv 13741 cnvtrcl0 37933 |
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