Theorem valcurfn1 14552

Description: The value of a curried function at O e. A is a partial application.

Hypothesis

Ref	Expression
valcurfn1.1	|- G = (F o. `'(2nd |` ({O} X. _V)))

Assertion

Ref	Expression
valcurfn1	|- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> ((cur1` F)` O) = G)

Proof of Theorem valcurfn1

Step	Hyp	Ref	Expression
1		fnex 4535	. . . . . . . . 9 |- ((F Fn (A X. B) /\ (A X. B) e. _V) -> F e. _V)
2	1	ex 402	. . . . . . . 8 |- (F Fn (A X. B) -> ((A X. B) e. _V -> F e. _V))
3		xpexg 4095	. . . . . . . 8 |- ((A e. C /\ B e. D) -> (A X. B) e. _V)
4	2, 3	syl5 20	. . . . . . 7 |- (F Fn (A X. B) -> ((A e. C /\ B e. D) -> F e. _V))
5	4	adantl 424	. . . . . 6 |- ((B =/= (/) /\ F Fn (A X. B)) -> ((A e. C /\ B e. D) -> F e. _V))
6	5	impcom 378	. . . . 5 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> F e. _V)
7	6	3adant3 896	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> F e. _V)
8		fnfun 4510	. . . . . 6 |- (F Fn (A X. B) -> Fun F)
9	8	adantl 424	. . . . 5 |- ((B =/= (/) /\ F Fn (A X. B)) -> Fun F)
10	9	3ad2ant2 898	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> Fun F)
11		fndm 4512	. . . . . . 7 |- (F Fn (A X. B) -> dom F = (A X. B))
12		relxp 4088	. . . . . . . 8 |- Rel (A X. B)
13		releq 4071	. . . . . . . 8 |- (dom F = (A X. B) -> (Rel dom F <-> Rel (A X. B)))
14	12, 13	mpbiri 211	. . . . . . 7 |- (dom F = (A X. B) -> Rel dom F)
15	11, 14	syl 12	. . . . . 6 |- (F Fn (A X. B) -> Rel dom F)
16	15	adantl 424	. . . . 5 |- ((B =/= (/) /\ F Fn (A X. B)) -> Rel dom F)
17	16	3ad2ant2 898	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> Rel dom F)
18		cur1val 14546	. . . 4 |- ((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	7, 10, 17, 18	syl111anc 1100	. . 3 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> (cur1` F) = {<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))})
20	19	fveq1d 4683	. 2 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> ((cur1` F)` O) = ({<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))}` O))
21		dmeq 4157	. . . . . . . . 9 |- (dom F = (A X. B) -> dom dom F = dom ( A X. B))
22		eqtr 1904	. . . . . . . . . . . 12 |- ((dom dom F = dom ( A X. B) /\ dom ( A X. B) = A) -> dom dom F = A)
23	22	eqcomd 1889	. . . . . . . . . . 11 |- ((dom dom F = dom ( A X. B) /\ dom ( A X. B) = A) -> A = dom dom F)
24	23	ex 402	. . . . . . . . . 10 |- (dom dom F = dom ( A X. B) -> (dom ( A X. B) = A -> A = dom dom F))
25		dmxp 4177	. . . . . . . . . 10 |- (B =/= (/) -> dom ( A X. B) = A)
26	24, 25	syl5 20	. . . . . . . . 9 |- (dom dom F = dom ( A X. B) -> (B =/= (/) -> A = dom dom F))
27	21, 26	syl 12	. . . . . . . 8 |- (dom F = (A X. B) -> (B =/= (/) -> A = dom dom F))
28	11, 27	syl 12	. . . . . . 7 |- (F Fn (A X. B) -> (B =/= (/) -> A = dom dom F))
29	28	impcom 378	. . . . . 6 |- ((B =/= (/) /\ F Fn (A X. B)) -> A = dom dom F)
30	29	eleq2d 1964	. . . . 5 |- ((B =/= (/) /\ F Fn (A X. B)) -> (O e. A <-> O e. dom dom F))
31	30	biimpa 460	. . . 4 |- (((B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> O e. dom dom F)
32	31	3adant1 894	. . 3 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> O e. dom dom F)
33		simprr 451	. . . . . . 7 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> F Fn (A X. B))
34		simpl 346	. . . . . . 7 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (A e. C /\ B e. D))
35	33, 34	jca 310	. . . . . 6 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (F Fn (A X. B) /\ (A e. C /\ B e. D)))
36	35	3adant3 896	. . . . 5 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> (F Fn (A X. B) /\ (A e. C /\ B e. D)))
37		domrancur1clem 14549	. . . . 5 |- ((F Fn (A X. B) /\ (A e. C /\ B e. D)) -> (F o. `'(2nd |` ({O} X. _V))) e. _V)
38	36, 37	syl 12	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> (F o. `'(2nd |` ({O} X. _V))) e. _V)
39		valcurfn1.1	. . . 4 |- G = (F o. `'(2nd |` ({O} X. _V)))
40	38, 39	syl5eqel 1975	. . 3 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> G e. _V)
41		sneq 3054	. . . . . . . . 9 |- (x = O -> {x} = {O})
42		xpeq1 4016	. . . . . . . . 9 |- ({x} = {O} -> ({x} X. _V) = ({O} X. _V))
43	41, 42	syl 12	. . . . . . . 8 |- (x = O -> ({x} X. _V) = ({O} X. _V))
44		reseq2 4219	. . . . . . . 8 |- (({x} X. _V) = ({O} X. _V) -> (2nd |` ({x} X. _V)) = (2nd |` ({O} X. _V)))
45	43, 44	syl 12	. . . . . . 7 |- (x = O -> (2nd |` ({x} X. _V)) = (2nd |` ({O} X. _V)))
46		cnveq 4135	. . . . . . 7 |- ((2nd |` ({x} X. _V)) = (2nd |` ({O} X. _V)) -> `'(2nd |` ({x} X. _V)) = `'(2nd |` ({O} X. _V)))
47	45, 46	syl 12	. . . . . 6 |- (x = O -> `'(2nd |` ({x} X. _V)) = `'(2nd |` ({O} X. _V)))
48	47	coeq2d 4128	. . . . 5 |- (x = O -> (F o. `'(2nd |` ({x} X. _V))) = (F o. `'(2nd |` ({O} X. _V))))
49	48, 39	syl6eqr 1946	. . . 4 |- (x = O -> (F o. `'(2nd |` ({x} X. _V))) = G)
50		eqid 1884	. . . 4 |- {<.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))))}
51	49, 50	fvopab4g 4742	. . 3 |- ((O e. dom dom F /\ G e. _V) -> ({<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))}` O) = G)
52	32, 40, 51	syl11anc 524	. 2 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> ({<.x, y>. | (x e. dom dom F /\ y = (F o. `'(2nd |` ({x} X. _V))))}` O) = G)
53	20, 52	eqtrd 1925	1 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B)) /\ O e. A) -> ((cur1` F)` O) = G)

Syntax hints:   -> wi 3   /\ 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:  valcurfn2 14553  gaplc 14731  gapm2 14732

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-f 4010  df-fo 4012  df-fv 4014  df-2nd 5021  df-cur1 14544

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