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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for funcestrcsetc 18042. (Contributed by AV, 22-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₯))) |
Ref | Expression |
---|---|
funcestrcsetclem1 | β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.f | . . 3 β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ π β π΅) β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
3 | fveq2 6843 | . . 3 β’ (π₯ = π β (Baseβπ₯) = (Baseβπ)) | |
4 | 3 | adantl 483 | . 2 β’ (((π β§ π β π΅) β§ π₯ = π) β (Baseβπ₯) = (Baseβπ)) |
5 | simpr 486 | . 2 β’ ((π β§ π β π΅) β π β π΅) | |
6 | fvexd 6858 | . 2 β’ ((π β§ π β π΅) β (Baseβπ) β V) | |
7 | 2, 4, 5, 6 | fvmptd 6956 | 1 β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 β¦ cmpt 5189 βcfv 6497 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-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-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: funcestrcsetclem2 18034 funcestrcsetclem7 18039 funcestrcsetclem8 18040 funcestrcsetclem9 18041 fullestrcsetc 18044 equivestrcsetc 18045 |
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