![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > setsidvald | Structured version Visualization version GIF version |
Description: Value of the structure
replacement function, deduction version.
Hint: Do not substitute π by a specific (positive) integer to be independent of a hard-coded index value. Often, (πΈβndx) can be used instead of π. (Contributed by AV, 14-Mar-2020.) (Revised by AV, 17-Oct-2024.) |
Ref | Expression |
---|---|
setsidvald.e | β’ πΈ = Slot π |
setsidvald.s | β’ (π β π β π) |
setsidvald.f | β’ (π β Fun π) |
setsidvald.d | β’ (π β π β dom π) |
Ref | Expression |
---|---|
setsidvald | β’ (π β π = (π sSet β¨π, (πΈβπ)β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsidvald.s | . . 3 β’ (π β π β π) | |
2 | fvex 6856 | . . 3 β’ (πΈβπ) β V | |
3 | setsval 17044 | . . 3 β’ ((π β π β§ (πΈβπ) β V) β (π sSet β¨π, (πΈβπ)β©) = ((π βΎ (V β {π})) βͺ {β¨π, (πΈβπ)β©})) | |
4 | 1, 2, 3 | sylancl 587 | . 2 β’ (π β (π sSet β¨π, (πΈβπ)β©) = ((π βΎ (V β {π})) βͺ {β¨π, (πΈβπ)β©})) |
5 | setsidvald.e | . . . . . 6 β’ πΈ = Slot π | |
6 | 5, 1 | strfvnd 17062 | . . . . 5 β’ (π β (πΈβπ) = (πβπ)) |
7 | 6 | opeq2d 4838 | . . . 4 β’ (π β β¨π, (πΈβπ)β© = β¨π, (πβπ)β©) |
8 | 7 | sneqd 4599 | . . 3 β’ (π β {β¨π, (πΈβπ)β©} = {β¨π, (πβπ)β©}) |
9 | 8 | uneq2d 4124 | . 2 β’ (π β ((π βΎ (V β {π})) βͺ {β¨π, (πΈβπ)β©}) = ((π βΎ (V β {π})) βͺ {β¨π, (πβπ)β©})) |
10 | setsidvald.f | . . 3 β’ (π β Fun π) | |
11 | setsidvald.d | . . 3 β’ (π β π β dom π) | |
12 | funresdfunsn 7136 | . . 3 β’ ((Fun π β§ π β dom π) β ((π βΎ (V β {π})) βͺ {β¨π, (πβπ)β©}) = π) | |
13 | 10, 11, 12 | syl2anc 585 | . 2 β’ (π β ((π βΎ (V β {π})) βͺ {β¨π, (πβπ)β©}) = π) |
14 | 4, 9, 13 | 3eqtrrd 2778 | 1 β’ (π β π = (π sSet β¨π, (πΈβπ)β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β cdif 3908 βͺ cun 3909 {csn 4587 β¨cop 4593 dom cdm 5634 βΎ cres 5636 Fun wfun 6491 βcfv 6497 (class class class)co 7358 sSet csts 17040 Slot cslot 17058 |
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 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-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 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-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-sets 17041 df-slot 17059 |
This theorem is referenced by: ressval3d 17132 |
Copyright terms: Public domain | W3C validator |