![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funcsetcestrclem4 | Structured version Visualization version GIF version |
Description: Lemma 4 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 |
---|---|
funcsetcestrclem4 | β’ (π β πΊ Fn (πΆ Γ πΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯))) = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯))) | |
2 | ovex 7391 | . . . 4 β’ (π¦ βm π₯) β V | |
3 | resiexg 7852 | . . . 4 β’ ((π¦ βm π₯) β V β ( I βΎ (π¦ βm π₯)) β V) | |
4 | 2, 3 | ax-mp 5 | . . 3 β’ ( I βΎ (π¦ βm π₯)) β V |
5 | 1, 4 | fnmpoi 8003 | . 2 β’ (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯))) Fn (πΆ Γ πΆ) |
6 | funcsetcestrc.g | . . 3 β’ (π β πΊ = (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯)))) | |
7 | 6 | fneq1d 6596 | . 2 β’ (π β (πΊ Fn (πΆ Γ πΆ) β (π₯ β πΆ, π¦ β πΆ β¦ ( I βΎ (π¦ βm π₯))) Fn (πΆ Γ πΆ))) |
8 | 5, 7 | mpbiri 258 | 1 β’ (π β πΊ Fn (πΆ Γ πΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 {csn 4587 β¨cop 4593 β¦ cmpt 5189 I cid 5531 Γ cxp 5632 βΎ cres 5636 Fn wfn 6492 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-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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-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-pw 4563 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-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: funcsetcestrc 18057 |
Copyright terms: Public domain | W3C validator |