![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem1ALTV | Structured version Visualization version GIF version |
Description: Lemma 1 for funcringcsetcALTV 46440. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funcringcsetcALTV.r | β’ π = (RingCatALTVβπ) |
funcringcsetcALTV.s | β’ π = (SetCatβπ) |
funcringcsetcALTV.b | β’ π΅ = (Baseβπ ) |
funcringcsetcALTV.c | β’ πΆ = (Baseβπ) |
funcringcsetcALTV.u | β’ (π β π β WUni) |
funcringcsetcALTV.f | β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
Ref | Expression |
---|---|
funcringcsetclem1ALTV | β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcringcsetcALTV.f | . . 3 β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ π β π΅) β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
3 | fveq2 6847 | . . 3 β’ (π₯ = π β (Baseβπ₯) = (Baseβπ)) | |
4 | 3 | adantl 483 | . 2 β’ (((π β§ π β π΅) β§ π₯ = π) β (Baseβπ₯) = (Baseβπ)) |
5 | simpr 486 | . 2 β’ ((π β§ π β π΅) β π β π΅) | |
6 | fvexd 6862 | . 2 β’ ((π β§ π β π΅) β (Baseβπ) β V) | |
7 | 2, 4, 5, 6 | fvmptd 6960 | 1 β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β¦ cmpt 5193 βcfv 6501 WUnicwun 10643 Basecbs 17090 SetCatcsetc 17968 RingCatALTVcringcALTV 46376 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 |
This theorem is referenced by: funcringcsetclem2ALTV 46432 funcringcsetclem7ALTV 46437 funcringcsetclem8ALTV 46438 funcringcsetclem9ALTV 46439 |
Copyright terms: Public domain | W3C validator |