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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem4ALTV | Structured version Visualization version GIF version |
Description: Lemma 4 for funcringcsetcALTV 47307. (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₯))) |
funcringcsetcALTV.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) |
Ref | Expression |
---|---|
funcringcsetclem4ALTV | β’ (π β πΊ Fn (π΅ Γ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦))) = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦))) | |
2 | ovex 7447 | . . . 4 β’ (π₯ RingHom π¦) β V | |
3 | id 22 | . . . . 5 β’ ((π₯ RingHom π¦) β V β (π₯ RingHom π¦) β V) | |
4 | 3 | resiexd 7222 | . . . 4 β’ ((π₯ RingHom π¦) β V β ( I βΎ (π₯ RingHom π¦)) β V) |
5 | 2, 4 | ax-mp 5 | . . 3 β’ ( I βΎ (π₯ RingHom π¦)) β V |
6 | 1, 5 | fnmpoi 8068 | . 2 β’ (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦))) Fn (π΅ Γ π΅) |
7 | funcringcsetcALTV.g | . . 3 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) | |
8 | 7 | fneq1d 6641 | . 2 β’ (π β (πΊ Fn (π΅ Γ π΅) β (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦))) Fn (π΅ Γ π΅))) |
9 | 6, 8 | mpbiri 258 | 1 β’ (π β πΊ Fn (π΅ Γ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3469 β¦ cmpt 5225 I cid 5569 Γ cxp 5670 βΎ cres 5674 Fn wfn 6537 βcfv 6542 (class class class)co 7414 β cmpo 7416 WUnicwun 10715 Basecbs 17171 SetCatcsetc 18055 RingHom crh 20397 RingCatALTVcringcALTV 47272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 |
This theorem is referenced by: funcringcsetcALTV 47307 |
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