![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funcestrcsetclem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcestrcsetc 18042. (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 |
---|---|
funcestrcsetclem5 | β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π βm π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.g | . . 3 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) |
3 | fveq2 6843 | . . . . . 6 β’ (π¦ = π β (Baseβπ¦) = (Baseβπ)) | |
4 | fveq2 6843 | . . . . . 6 β’ (π₯ = π β (Baseβπ₯) = (Baseβπ)) | |
5 | 3, 4 | oveqan12rd 7378 | . . . . 5 β’ ((π₯ = π β§ π¦ = π) β ((Baseβπ¦) βm (Baseβπ₯)) = ((Baseβπ) βm (Baseβπ))) |
6 | funcestrcsetc.n | . . . . . 6 β’ π = (Baseβπ) | |
7 | funcestrcsetc.m | . . . . . 6 β’ π = (Baseβπ) | |
8 | 6, 7 | oveq12i 7370 | . . . . 5 β’ (π βm π) = ((Baseβπ) βm (Baseβπ)) |
9 | 5, 8 | eqtr4di 2791 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β ((Baseβπ¦) βm (Baseβπ₯)) = (π βm π)) |
10 | 9 | reseq2d 5938 | . . 3 β’ ((π₯ = π β§ π¦ = π) β ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))) = ( I βΎ (π βm π))) |
11 | 10 | adantl 483 | . 2 β’ (((π β§ (π β π΅ β§ π β π΅)) β§ (π₯ = π β§ π¦ = π)) β ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))) = ( I βΎ (π βm π))) |
12 | simprl 770 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
13 | simprr 772 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
14 | ovexd 7393 | . . 3 β’ ((π β§ (π β π΅ β§ π β π΅)) β (π βm π) β V) | |
15 | 14 | resiexd 7167 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β ( I βΎ (π βm π)) β V) |
16 | 2, 11, 12, 13, 15 | ovmpod 7508 | 1 β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π βm π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 β¦ cmpt 5189 I cid 5531 βΎ cres 5636 βcfv 6497 (class class class)co 7358 β cmpo 7360 βm cmap 8768 WUnicwun 10641 Basecbs 17088 SetCatcsetc 17966 ExtStrCatcestrc 18014 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 |
This theorem is referenced by: funcestrcsetclem6 18038 funcestrcsetclem7 18039 funcestrcsetclem8 18040 funcestrcsetclem9 18041 |
Copyright terms: Public domain | W3C validator |