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Mirrors > Home > MPE Home > Th. List > lmimfn | Structured version Visualization version GIF version |
Description: Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
Ref | Expression |
---|---|
lmimfn | ⊢ LMIso Fn (LMod × LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lmim 20622 | . 2 ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) | |
2 | ovex 7437 | . . 3 ⊢ (𝑠 LMHom 𝑡) ∈ V | |
3 | 2 | rabex 5331 | . 2 ⊢ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V |
4 | 1, 3 | fnmpoi 8051 | 1 ⊢ LMIso Fn (LMod × LMod) |
Colors of variables: wff setvar class |
Syntax hints: {crab 3433 × cxp 5673 Fn wfn 6535 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 LModclmod 20459 LMHom clmhm 20618 LMIso clmim 20619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-lmim 20622 |
This theorem is referenced by: brlmic 20667 |
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