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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 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©})) |
Ref | Expression |
---|---|
funcsetcestrclem1 | β’ ((π β§ π β πΆ) β (πΉβπ) = {β¨(Baseβndx), πβ©}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.f | . . 3 β’ (π β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ π β πΆ) β πΉ = (π₯ β πΆ β¦ {β¨(Baseβndx), π₯β©})) |
3 | opeq2 4832 | . . . 4 β’ (π₯ = π β β¨(Baseβndx), π₯β© = β¨(Baseβndx), πβ©) | |
4 | 3 | sneqd 4599 | . . 3 β’ (π₯ = π β {β¨(Baseβndx), π₯β©} = {β¨(Baseβndx), πβ©}) |
5 | 4 | adantl 483 | . 2 β’ (((π β§ π β πΆ) β§ π₯ = π) β {β¨(Baseβndx), π₯β©} = {β¨(Baseβndx), πβ©}) |
6 | simpr 486 | . 2 β’ ((π β§ π β πΆ) β π β πΆ) | |
7 | snex 5389 | . . 3 β’ {β¨(Baseβndx), πβ©} β V | |
8 | 7 | a1i 11 | . 2 β’ ((π β§ π β πΆ) β {β¨(Baseβndx), πβ©} β V) |
9 | 2, 5, 6, 8 | fvmptd 6956 | 1 β’ ((π β§ π β πΆ) β (πΉβπ) = {β¨(Baseβndx), πβ©}) |
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 βcfv 6497 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-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-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-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 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-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: funcsetcestrclem2 18048 embedsetcestrclem 18050 funcsetcestrclem7 18054 funcsetcestrclem8 18055 funcsetcestrclem9 18056 fullsetcestrc 18059 |
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