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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 6898 | . . 3 β’ (πΈβπ) β V | |
3 | setsval 17109 | . . 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 585 | . 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 17127 | . . . . 5 β’ (π β (πΈβπ) = (πβπ)) |
7 | 6 | opeq2d 4875 | . . . 4 β’ (π β β¨π, (πΈβπ)β© = β¨π, (πβπ)β©) |
8 | 7 | sneqd 4635 | . . 3 β’ (π β {β¨π, (πΈβπ)β©} = {β¨π, (πβπ)β©}) |
9 | 8 | uneq2d 4158 | . 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 7183 | . . 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 583 | . 2 β’ (π β ((π βΎ (V β {π})) βͺ {β¨π, (πβπ)β©}) = π) |
14 | 4, 9, 13 | 3eqtrrd 2771 | 1 β’ (π β π = (π sSet β¨π, (πΈβπ)β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β cdif 3940 βͺ cun 3941 {csn 4623 β¨cop 4629 dom cdm 5669 βΎ cres 5671 Fun wfun 6531 βcfv 6537 (class class class)co 7405 sSet csts 17105 Slot cslot 17123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-sets 17106 df-slot 17124 |
This theorem is referenced by: ressval3d 17200 opprabs 33102 |
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