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Mirrors > Home > MPE Home > Th. List > frlmrcl | Structured version Visualization version GIF version |
Description: If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlmrcl.b | ⊢ 𝐵 = (Base‘𝐹) |
Ref | Expression |
---|---|
frlmrcl | ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmval.f | . 2 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | frlmrcl.b | . 2 ⊢ 𝐵 = (Base‘𝐹) | |
3 | df-frlm 20885 | . . 3 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | |
4 | 3 | reldmmpo 7279 | . 2 ⊢ Rel dom freeLMod |
5 | 1, 2, 4 | strov2rcl 16540 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 {csn 4561 × cxp 5548 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 ringLModcrglmod 19935 ⊕m cdsmm 20869 freeLMod cfrlm 20884 |
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-oprab 7154 df-mpo 7155 df-slot 16481 df-base 16483 df-frlm 20885 |
This theorem is referenced by: frlmbasfsupp 20896 frlmbasmap 20897 frlmvscafval 20904 |
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