Theorem xpmapenlem3 3393

Description: Lemma for xpmapen 3396.

Hypotheses

Ref	Expression
xpmapen.1	|- A e. V
xpmapen.2	|- B e. V
xpmapen.3	|- C e. V
xpmapenlem.4	|- D = {<.z, w>. | (z e. C /\ w = U.dom {(x` z)})}
xpmapenlem.5	|- R = {<.z, w>. | (z e. C /\ w = U.ran {(x` z)})}
xpmapenlem.6	|- S = {<.z, w>. | (z e. C /\ w = <.(U.dom {y}` z), (U.ran {y}` z)>.)}

Assertion

Ref	Expression
xpmapenlem3	|- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = S)

Distinct variable group(s):   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   y,D   y,R   x,S

Proof of Theorem xpmapenlem3

Step	Hyp	Ref	Expression
1		ffn 2752	. . . 4 |- (x:C-->(A X. B) -> x Fn C)
2		fnopabfv 2858	. . . 4 |- (x Fn C <-> x = {<.z, w>. | (z e. C /\ w = (x` z))})
3	1, 2	sylib 173	. . 3 |- (x:C-->(A X. B) -> x = {<.z, w>. | (z e. C /\ w = (x` z))})
4	3	adantr 306	. 2 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = {<.z, w>. | (z e. C /\ w = (x` z))})
5		ax-17 925	. . . . 5 |- (x:C-->(A X. B) -> A.z x:C-->(A X. B))
6		xpmapen.1	. . . . . . 7 |- A e. V
7		xpmapen.2	. . . . . . 7 |- B e. V
8		xpmapen.3	. . . . . . 7 |- C e. V
9		xpmapenlem.4	. . . . . . 7 |- D = {<.z, w>. | (z e. C /\ w = U.dom {(x` z)})}
10		xpmapenlem.5	. . . . . . 7 |- R = {<.z, w>. | (z e. C /\ w = U.ran {(x` z)})}
11		xpmapenlem.6	. . . . . . 7 |- S = {<.z, w>. | (z e. C /\ w = <.(U.dom {y}` z), (U.ran {y}` z)>.)}
12	6, 7, 8, 9, 10, 11	xpmapenlem1 3391	. . . . . 6 |- ((y = <.D, R>. -> A.z y = <.D, R>.) /\ (y = <.D, R>. -> A.w y = <.D, R>.))
13	12	pm3.26i 257	. . . . 5 |- (y = <.D, R>. -> A.z y = <.D, R>.)
14	5, 13	hban 704	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.z(x:C-->(A X. B) /\ y = <.D, R>.))
15		ax-17 925	. . . . 5 |- (x:C-->(A X. B) -> A.w x:C-->(A X. B))
16	12	pm3.27i 261	. . . . 5 |- (y = <.D, R>. -> A.w y = <.D, R>.)
17	15, 16	hban 704	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.w(x:C-->(A X. B) /\ y = <.D, R>.))
18		ffvrn 2890	. . . . . . . . . 10 |- ((x:C-->(A X. B) /\ z e. C) -> (x` z) e. (A X. B))
19		elxp4 2640	. . . . . . . . . . 11 |- ((x` z) e. (A X. B) <-> ((x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>. /\ (U.dom {(x` z)} e. A /\ U.ran {(x` z)} e. B)))
20	19	pm3.26bd 259	. . . . . . . . . 10 |- ((x` z) e. (A X. B) -> (x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
21	18, 20	syl 12	. . . . . . . . 9 |- ((x:C-->(A X. B) /\ z e. C) -> (x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
22	21	adantlr 310	. . . . . . . 8 |- (((x:C-->(A X. B) /\ y = <.D, R>.) /\ z e. C) -> (x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
23	6, 6, 8, 9, 10, 11	xpmapenlem2 3392	. . . . . . . . . 10 |- ((y = <.D, R>. /\ z e. C) -> ((U.dom {y}` z) = U.dom {(x` z)} /\ (U.ran {y}` z) = U.ran {(x` z)}))
24		opeq12 1878	. . . . . . . . . 10 |- (((U.dom {y}` z) = U.dom {(x` z)} /\ (U.ran {y}` z) = U.ran {(x` z)}) -> <.(U.dom {y}` z), (U.ran {y}` z)>. = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
25	23, 24	syl 12	. . . . . . . . 9 |- ((y = <.D, R>. /\ z e. C) -> <.(U.dom {y}` z), (U.ran {y}` z)>. = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
26	25	adantll 309	. . . . . . . 8 |- (((x:C-->(A X. B) /\ y = <.D, R>.) /\ z e. C) -> <.(U.dom {y}` z), (U.ran {y}` z)>. = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
27	22, 26	eqtr4d 1131	. . . . . . 7 |- (((x:C-->(A X. B) /\ y = <.D, R>.) /\ z e. C) -> (x` z) = <.(U.dom {y}` z), (U.ran {y}` z)>.)
28	27	cleq2d 1112	. . . . . 6 |- (((x:C-->(A X. B) /\ y = <.D, R>.) /\ z e. C) -> (w = (x` z) <-> w = <.(U.dom {y}` z), (U.ran {y}` z)>.))
29	28	exp 291	. . . . 5 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> (z e. C -> (w = (x` z) <-> w = <.(U.dom {y}` z), (U.ran {y}` z)>.)))
30	29	pm5.32d 491	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> ((z e. C /\ w = (x` z)) <-> (z e. C /\ w = <.(U.dom {y}` z), (U.ran {y}` z)>.)))
31	14, 17, 30	biopabd 2101	. . 3 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> {<.z, w>. | (z e. C /\ w = (x` z))} = {<.z, w>. | (z e. C /\ w = <.(U.dom {y}` z), (U.ran {y}` z)>.)})
32	31, 11	syl6eqr 1142	. 2 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> {<.z, w>. | (z e. C /\ w = (x` z))} = S)
33	4, 32	eqtrd 1128	1 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = S)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  U.cuni 1919  {copab 2055   X. cxp 2408  dom cdm 2410  ran crn 2411   Fn wfn 2417  -->wf 2418  ` cfv 2422

This theorem is referenced by:  xpmapenlem5 3395

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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