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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 7194 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅) |
5 | 2, 4 | syl5eq 2868 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅) |
6 | 5 | fveq2d 6674 | . . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅)) |
7 | elbasov.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
8 | base0 16536 | . . 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 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 dom cdm 5555 Rel wrel 5560 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 |
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-pow 5266 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-sbc 3773 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-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-slot 16487 df-base 16489 |
This theorem is referenced by: strov2rcl 16546 psrelbas 20159 psraddcl 20163 psrmulcllem 20167 psrvscafval 20170 psrvscacl 20173 resspsradd 20196 resspsrmul 20197 cphsubrglem 23781 mdegcl 24663 |
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