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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem6 | Structured version Visualization version GIF version |
Description: Lemma 6 for funcestrcsetc 18131. (Contributed by AV, 23-Mar-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | β’ πΈ = (ExtStrCatβπ) |
funcestrcsetc.s | β’ π = (SetCatβπ) |
funcestrcsetc.b | β’ π΅ = (BaseβπΈ) |
funcestrcsetc.c | β’ πΆ = (Baseβπ) |
funcestrcsetc.u | β’ (π β π β WUni) |
funcestrcsetc.f | β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
funcestrcsetc.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) |
funcestrcsetc.m | β’ π = (Baseβπ) |
funcestrcsetc.n | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
funcestrcsetclem6 | β’ ((π β§ (π β π΅ β§ π β π΅) β§ π» β (π βm π)) β ((ππΊπ)βπ») = π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.e | . . . . 5 β’ πΈ = (ExtStrCatβπ) | |
2 | funcestrcsetc.s | . . . . 5 β’ π = (SetCatβπ) | |
3 | funcestrcsetc.b | . . . . 5 β’ π΅ = (BaseβπΈ) | |
4 | funcestrcsetc.c | . . . . 5 β’ πΆ = (Baseβπ) | |
5 | funcestrcsetc.u | . . . . 5 β’ (π β π β WUni) | |
6 | funcestrcsetc.f | . . . . 5 β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) | |
7 | funcestrcsetc.g | . . . . 5 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) | |
8 | funcestrcsetc.m | . . . . 5 β’ π = (Baseβπ) | |
9 | funcestrcsetc.n | . . . . 5 β’ π = (Baseβπ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | funcestrcsetclem5 18126 | . . . 4 β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π βm π))) |
11 | 10 | 3adant3 1130 | . . 3 β’ ((π β§ (π β π΅ β§ π β π΅) β§ π» β (π βm π)) β (ππΊπ) = ( I βΎ (π βm π))) |
12 | 11 | fveq1d 6893 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅) β§ π» β (π βm π)) β ((ππΊπ)βπ») = (( I βΎ (π βm π))βπ»)) |
13 | fvresi 7176 | . . 3 β’ (π» β (π βm π) β (( I βΎ (π βm π))βπ») = π») | |
14 | 13 | 3ad2ant3 1133 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅) β§ π» β (π βm π)) β (( I βΎ (π βm π))βπ») = π») |
15 | 12, 14 | eqtrd 2767 | 1 β’ ((π β§ (π β π΅ β§ π β π΅) β§ π» β (π βm π)) β ((ππΊπ)βπ») = π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β¦ cmpt 5225 I cid 5569 βΎ cres 5674 βcfv 6542 (class class class)co 7414 β cmpo 7416 βm cmap 8836 WUnicwun 10715 Basecbs 17171 SetCatcsetc 18055 ExtStrCatcestrc 18103 |
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 |
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 |
This theorem is referenced by: funcestrcsetclem9 18130 fthestrcsetc 18132 fullestrcsetc 18133 |
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