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Mirrors > Home > MPE Home > Th. List > strfv2 | Structured version Visualization version GIF version |
Description: A variation on strfv 17081 to avoid asserting that π itself is a function, which involves sethood of all the ordered pair components of π. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
strfv2.s | β’ π β V |
strfv2.f | β’ Fun β‘β‘π |
strfv2.e | β’ πΈ = Slot (πΈβndx) |
strfv2.n | β’ β¨(πΈβndx), πΆβ© β π |
Ref | Expression |
---|---|
strfv2 | β’ (πΆ β π β πΆ = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfv2.e | . 2 β’ πΈ = Slot (πΈβndx) | |
2 | strfv2.s | . . 3 β’ π β V | |
3 | 2 | a1i 11 | . 2 β’ (πΆ β π β π β V) |
4 | strfv2.f | . . 3 β’ Fun β‘β‘π | |
5 | 4 | a1i 11 | . 2 β’ (πΆ β π β Fun β‘β‘π) |
6 | strfv2.n | . . 3 β’ β¨(πΈβndx), πΆβ© β π | |
7 | 6 | a1i 11 | . 2 β’ (πΆ β π β β¨(πΈβndx), πΆβ© β π) |
8 | id 22 | . 2 β’ (πΆ β π β πΆ β π) | |
9 | 1, 3, 5, 7, 8 | strfv2d 17079 | 1 β’ (πΆ β π β πΆ = (πΈβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β¨cop 4593 β‘ccnv 5633 Fun wfun 6491 βcfv 6497 Slot cslot 17058 ndxcnx 17070 |
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-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-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-slot 17059 |
This theorem is referenced by: strfv 17081 |
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