Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelref | Structured version Visualization version GIF version |
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) |
Ref | Expression |
---|---|
eqvrelref.1 | ⊢ (𝜑 → EqvRel 𝑅) |
eqvrelref.2 | ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) |
Ref | Expression |
---|---|
eqvrelref | ⊢ (𝜑 → 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelref.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) | |
2 | eqvrelref.1 | . . . 4 ⊢ (𝜑 → EqvRel 𝑅) | |
3 | eqvrelrel 35847 | . . . 4 ⊢ ( EqvRel 𝑅 → Rel 𝑅) | |
4 | releldmb 5816 | . . . 4 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) |
6 | 1, 5 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑥 𝐴𝑅𝑥) |
7 | 2 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → EqvRel 𝑅) |
8 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝑥) | |
9 | 7, 8, 8 | eqvreltr4d 35859 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝑥) → 𝐴𝑅𝐴) |
10 | 6, 9 | exlimddv 1936 | 1 ⊢ (𝜑 → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 class class class wbr 5066 dom cdm 5555 Rel wrel 5560 EqvRel weqvrel 35485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-refrel 35767 df-symrel 35795 df-trrel 35825 df-eqvrel 35835 |
This theorem is referenced by: eqvrelth 35861 |
Copyright terms: Public domain | W3C validator |