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| Mirrors > Home > MPE Home > Th. List > erex | Structured version Visualization version GIF version | ||
| Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| erex | ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | erssxp 8768 | . . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴)) | |
| 2 | sqxpexg 7775 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
| 3 | ssexg 5323 | . . 3 ⊢ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝑅 Er 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ V) | 
| 5 | 4 | ex 412 | 1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 × cxp 5683 Er wer 8742 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-er 8745 | 
| This theorem is referenced by: erexb 8770 qliftlem 8838 qshash 15863 qusaddvallem 17596 qusaddflem 17597 qusaddval 17598 qusaddf 17599 qusmulval 17600 qusmulf 17601 qusgrp2 19076 efgrelexlemb 19768 efgcpbllemb 19773 frgpuplem 19790 qusrng 20177 qusring2 20331 vitalilem2 25644 vitalilem3 25645 tgjustr 28482 | 
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