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Mirrors > Home > MPE Home > Th. List > strov2rcl | Structured version Visualization version GIF version |
Description: Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.) |
Ref | Expression |
---|---|
strov2rcl.s | ⊢ 𝑆 = (𝐼𝐹𝑅) |
strov2rcl.b | ⊢ 𝐵 = (Base‘𝑆) |
strov2rcl.f | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
strov2rcl | ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strov2rcl.f | . . 3 ⊢ Rel dom 𝐹 | |
2 | strov2rcl.s | . . 3 ⊢ 𝑆 = (𝐼𝐹𝑅) | |
3 | strov2rcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
4 | 1, 2, 3 | elbasov 16537 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
5 | 4 | simpld 498 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 dom cdm 5519 Rel wrel 5524 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-slot 16479 df-base 16481 |
This theorem is referenced by: dsmmbas2 20426 frlmrcl 20446 mplrcl 20729 psropprmul 20867 |
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