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Mirrors > Home > MPE Home > Th. List > funcestrcsetclem4 | Structured version Visualization version GIF version |
Description: Lemma 4 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₯))) |
funcestrcsetc.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) |
Ref | Expression |
---|---|
funcestrcsetclem4 | β’ (π β πΊ Fn (π΅ Γ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯)))) = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯)))) | |
2 | ovex 7391 | . . . 4 β’ ((Baseβπ¦) βm (Baseβπ₯)) β V | |
3 | resiexg 7852 | . . . 4 β’ (((Baseβπ¦) βm (Baseβπ₯)) β V β ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))) β V) | |
4 | 2, 3 | ax-mp 5 | . . 3 β’ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))) β V |
5 | 1, 4 | fnmpoi 8003 | . 2 β’ (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯)))) Fn (π΅ Γ π΅) |
6 | funcestrcsetc.g | . . 3 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯))))) | |
7 | 6 | fneq1d 6596 | . 2 β’ (π β (πΊ Fn (π΅ Γ π΅) β (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ ((Baseβπ¦) βm (Baseβπ₯)))) Fn (π΅ Γ π΅))) |
8 | 5, 7 | mpbiri 258 | 1 β’ (π β πΊ Fn (π΅ Γ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 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 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-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: funcestrcsetc 18042 |
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