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Mirrors > Home > MPE Home > Th. List > elbasov | Structured version Visualization version GIF version |
Description: Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
elbasov.o | ⊢ Rel dom 𝑂 |
elbasov.s | ⊢ 𝑆 = (𝑋𝑂𝑌) |
elbasov.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
elbasov | ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4299 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | elbasov.s | . . . . 5 ⊢ 𝑆 = (𝑋𝑂𝑌) | |
3 | elbasov.o | . . . . . 6 ⊢ Rel dom 𝑂 | |
4 | 3 | ovprc 7188 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅) |
5 | 2, 4 | syl5eq 2868 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅) |
6 | 5 | fveq2d 6669 | . . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅)) |
7 | elbasov.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
8 | base0 16530 | . . 3 ⊢ ∅ = (Base‘∅) | |
9 | 6, 7, 8 | 3eqtr4g 2881 | . 2 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅) |
10 | 1, 9 | nsyl2 143 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 dom cdm 5550 Rel wrel 5555 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-slot 16481 df-base 16483 |
This theorem is referenced by: strov2rcl 16540 psrelbas 20153 psraddcl 20157 psrmulcllem 20161 psrvscafval 20164 psrvscacl 20167 resspsradd 20190 resspsrmul 20191 cphsubrglem 23775 mdegcl 24657 |
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