Theorem cur1val 14546

Description: The value of a curried operation.

Assertion

Ref	Expression
cur1val	|- ((F e. A /\ Fun F /\ Rel dom F) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})

Distinct variable group:   x,F,y

Proof of Theorem cur1val

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (F e. A -> F e. _V)
2	1	3ad2ant1 897	. 2 |- ((F e. A /\ Fun F /\ Rel dom F) -> F e. _V)
3		dmexg 4206	. . . . 5 |- (F e. A -> dom F e. _V)
4		dmexg 4206	. . . . 5 |- (dom F e. _V -> dom dom F e. _V)
5	3, 4	syl 12	. . . 4 |- (F e. A -> dom dom F e. _V)
6	5	3ad2ant1 897	. . 3 |- ((F e. A /\ Fun F /\ Rel dom F) -> dom dom F e. _V)
7		opabex2g 4540	. . 3 |- (dom dom F e. _V -> {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))} e. _V)
8	6, 7	syl 12	. 2 |- ((F e. A /\ Fun F /\ Rel dom F) -> {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))} e. _V)
9		3simpc 874	. 2 |- ((F e. A /\ Fun F /\ Rel dom F) -> (Fun F /\ Rel dom F))
10		df-cur1 14544	. . 3 |- cur1 = {<.f, g>. | ((Fun f /\ Rel dom f) /\ g = {<.x, y>. | (x e. dom dom f /\ y = (f o. `'(2nd |` ({x} X. _V))))})}
11		funeq 4441	. . . 4 |- (f = F -> (Fun f <-> Fun F))
12		dmeq 4157	. . . . 5 |- (f = F -> dom f = dom F)
13		releq 4071	. . . . 5 |- (dom f = dom F -> (Rel dom f <-> Rel dom F))
14	12, 13	syl 12	. . . 4 |- (f = F -> (Rel dom f <-> Rel dom F))
15	11, 14	anbi12d 690	. . 3 |- (f = F -> ((Fun f /\ Rel dom f) <-> (Fun F /\ Rel dom F)))
16	12	dmeqd 4159	. . . . . 6 |- (f = F -> dom dom f = dom dom F)
17	16	eleq2d 1964	. . . . 5 |- (f = F -> (x e. dom dom f <-> x e. dom dom F))
18		coeq1 4123	. . . . . 6 |- (f = F -> (f o. `'(2nd |` ({x} X. _V))) = (F o. `'(2nd |` ({x} X. _V))))
19	18	eqeq2d 1895	. . . . 5 |- (f = F -> (y = (f o. `'(2nd |` ({x} X. _V))) <-> y = (F o. `'(2nd |` ({x} X. _V)))))
20	17, 19	anbi12d 690	. . . 4 |- (f = F -> ((x e. dom dom f /\ y = (f o. `'(2nd |` ({x} X. _V)))) <-> (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))))
21	20	opabbidv 3401	. . 3 |- (f = F -> {<.x, y>. | (x e. dom dom f /\ y = (f o. `'(2nd |` ({x} X. _V))))} = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
22	10, 15, 21	fvopab6 4757	. 2 |- ((F e. _V /\ {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))} e. _V /\ (Fun F /\ Rel dom F)) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
23	2, 8, 9, 22	syl111anc 1100	1 |- ((F e. A /\ Fun F /\ Rel dom F) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986   |` cres 3988   o. ccom 3990  Rel wrel 3991  Fun wfun 3992  ` cfv 3998  2ndc2nd 5019  cur1ccur1 14542

This theorem is referenced by:  cur1vald 14547  valcurfn1 14552

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-cur1 14544

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