Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gimfn | Structured version Visualization version GIF version |
Description: The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
gimfn | ⊢ GrpIso Fn (Grp × Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-gim 17701 | . 2 ⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) | |
2 | ovex 6678 | . . 3 ⊢ (𝑠 GrpHom 𝑡) ∈ V | |
3 | 2 | rabex 4813 | . 2 ⊢ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V |
4 | 1, 3 | fnmpt2i 7239 | 1 ⊢ GrpIso Fn (Grp × Grp) |
Colors of variables: wff setvar class |
Syntax hints: {crab 2916 × cxp 5112 Fn wfn 5883 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Grpcgrp 17422 GrpHom cghm 17657 GrpIso cgim 17699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-gim 17701 |
This theorem is referenced by: brgic 17711 gicer 17718 gicerOLD 17719 |
Copyright terms: Public domain | W3C validator |