Theorem cur1vald 14547

Description: The value of a curried operation.

Assertion

Ref	Expression
cur1vald	|- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> (cur1` F) = {<.x, y>. | (x e. A /\ y = (F o. `'(2nd |` ({x} X. _V))))})

Distinct variable groups:   x,A,y   x,B,y   x,C,y   x,D,y   x,F,y

Proof of Theorem cur1vald

Step	Hyp	Ref	Expression
1		simp1 876	. . . . . . 7 |- ((F Fn (A X. B) /\ A e. C /\ B e. D) -> F Fn (A X. B))
2		xpexg 4095	. . . . . . . 8 |- ((A e. C /\ B e. D) -> (A X. B) e. _V)
3	2	3adant1 894	. . . . . . 7 |- ((F Fn (A X. B) /\ A e. C /\ B e. D) -> (A X. B) e. _V)
4		fnex 4535	. . . . . . 7 |- ((F Fn (A X. B) /\ (A X. B) e. _V) -> F e. _V)
5	1, 3, 4	syl11anc 524	. . . . . 6 |- ((F Fn (A X. B) /\ A e. C /\ B e. D) -> F e. _V)
6	5	3expib 1070	. . . . 5 |- (F Fn (A X. B) -> ((A e. C /\ B e. D) -> F e. _V))
7	6	adantr 425	. . . 4 |- ((F Fn (A X. B) /\ B =/= (/)) -> ((A e. C /\ B e. D) -> F e. _V))
8	7	imp 377	. . 3 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> F e. _V)
9		fnfun 4510	. . . 4 |- (F Fn (A X. B) -> Fun F)
10	9	ad2antrr 440	. . 3 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> Fun F)
11		fndm 4512	. . . . 5 |- (F Fn (A X. B) -> dom F = (A X. B))
12		relxp 4088	. . . . . 6 |- Rel (A X. B)
13		releq 4071	. . . . . . 7 |- ((A X. B) = dom F -> (Rel (A X. B) <-> Rel dom F))
14	13	eqcoms 1887	. . . . . 6 |- (dom F = (A X. B) -> (Rel (A X. B) <-> Rel dom F))
15	12, 14	mpbii 210	. . . . 5 |- (dom F = (A X. B) -> Rel dom F)
16	11, 15	syl 12	. . . 4 |- (F Fn (A X. B) -> Rel dom F)
17	16	ad2antrr 440	. . 3 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> Rel dom F)
18		cur1val 14546	. . 3 |- ((F e. _V /\ Fun F /\ Rel dom F) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
19	8, 10, 17, 18	syl111anc 1100	. 2 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
20		dmeq 4157	. . . . . . . . 9 |- (dom F = (A X. B) -> dom dom F = dom ( A X. B))
21		dmxp 4177	. . . . . . . . . . . 12 |- (B =/= (/) -> dom ( A X. B) = A)
22	21	eqeq1d 1892	. . . . . . . . . . 11 |- (B =/= (/) -> (dom ( A X. B) = dom dom F <-> A = dom dom F))
23	22	biimpcd 172	. . . . . . . . . 10 |- (dom ( A X. B) = dom dom F -> (B =/= (/) -> A = dom dom F))
24	23	eqcoms 1887	. . . . . . . . 9 |- (dom dom F = dom ( A X. B) -> (B =/= (/) -> A = dom dom F))
25	20, 24	syl 12	. . . . . . . 8 |- (dom F = (A X. B) -> (B =/= (/) -> A = dom dom F))
26	11, 25	syl 12	. . . . . . 7 |- (F Fn (A X. B) -> (B =/= (/) -> A = dom dom F))
27	26	imp 377	. . . . . 6 |- ((F Fn (A X. B) /\ B =/= (/)) -> A = dom dom F)
28	27	adantr 425	. . . . 5 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> A = dom dom F)
29	28	eleq2d 1964	. . . 4 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> (x e. A <-> x e. dom dom F))
30	29	anbi1d 679	. . 3 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> ((x e. A /\ y = (F o. `'(2nd |` ({x} X. _V)))) <-> (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))))
31	30	opabbidv 3401	. 2 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> {<.x, y>. | (x e. A /\ y = (F o. `'(2nd |` ({x} X. _V))))} = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
32	19, 31	eqtr4d 1928	1 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> (cur1` F) = {<.x, y>. | (x e. A /\ y = (F o. `'(2nd |` ({x} X. _V))))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  {csn 3044  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986   |` cres 3988   o. ccom 3990  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  2ndc2nd 5019  cur1ccur1 14542

This theorem is referenced by:  domrancur1b 14548  domrancur1c 14550

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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