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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcsetcestrc 18057. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | β’ π = (SetCatβπ) |
funcsetcestrc.c | β’ πΆ = (Baseβπ) |
funcsetcestrc.f | β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) |
funcsetcestrc.u | β’ (π β π β WUni) |
funcsetcestrc.o | β’ (π β Ο β π) |
funcsetcestrc.g | β’ (π β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) |
Ref | Expression |
---|---|
funcsetcestrclem5 | β’ ((π β§ (π β πΆ β§ π β πΆ)) β (ππΊπ) = ( I βΎ (π βm π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.g | . . 3 β’ (π β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ)) β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) |
3 | oveq12 7367 | . . . . 5 β’ ((π¦ = π β§ π₯ = π) β (π¦ βm π₯) = (π βm π)) | |
4 | 3 | ancoms 460 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β (π¦ βm π₯) = (π βm π)) |
5 | 4 | reseq2d 5938 | . . 3 β’ ((π₯ = π β§ π¦ = π) β ( I βΎ (π¦ βm π₯)) = ( I βΎ (π βm π))) |
6 | 5 | adantl 483 | . 2 β’ (((π β§ (π β πΆ β§ π β πΆ)) β§ (π₯ = π β§ π¦ = π)) β ( I βΎ (π¦ βm π₯)) = ( I βΎ (π βm π))) |
7 | simprl 770 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ)) β π β πΆ) | |
8 | simprr 772 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ)) β π β πΆ) | |
9 | ovexd 7393 | . . 3 β’ ((π β§ (π β πΆ β§ π β πΆ)) β (π βm π) β V) | |
10 | 9 | resiexd 7167 | . 2 β’ ((π β§ (π β πΆ β§ π β πΆ)) β ( I βΎ (π βm π)) β V) |
11 | 2, 6, 7, 8, 10 | ovmpod 7508 | 1 β’ ((π β§ (π β πΆ β§ π β πΆ)) 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 {csn 4587 β¨cop 4593 β¦ cmpt 5189 I cid 5531 βΎ cres 5636 βcfv 6497 (class class class)co 7358 β cmpo 7360 Οcom 7803 βm cmap 8768 WUnicwun 10641 ndxcnx 17070 Basecbs 17088 SetCatcsetc 17966 |
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: funcsetcestrclem6 18053 funcsetcestrclem7 18054 funcsetcestrclem8 18055 funcsetcestrclem9 18056 |
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