Theorem xpmapenlem3 4498

Description: Lemma for xpmapen 4501.

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 groups:   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 3627	. . . 4 |- (x:C-->(A X. B) -> x Fn C)
2		fnopabfv 3758	. . . 4 |- (x Fn C <-> x = {<.z, w>. | (z e. C /\ w = (x` z))})
3	1, 2	sylib 198	. . 3 |- (x:C-->(A X. B) -> x = {<.z, w>. | (z e. C /\ w = (x` z))})
4	3	adantr 389	. 2 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = {<.z, w>. | (z e. C /\ w = (x` z))})
5		ax-17 971	. . . . 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 4496	. . . . . 6 |- ((y = <.D, R>. -> A.z y = <.D, R>.) /\ (y = <.D, R>. -> A.w y = <.D, R>.))
13	12	pm3.26i 320	. . . . 5 |- (y = <.D, R>. -> A.z y = <.D, R>.)
14	5, 13	hban 1009	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.z(x:C-->(A X. B) /\ y = <.D, R>.))
15		ax-17 971	. . . . 5 |- (x:C-->(A X. B) -> A.w x:C-->(A X. B))
16	12	pm3.27i 324	. . . . 5 |- (y = <.D, R>. -> A.w y = <.D, R>.)
17	15, 16	hban 1009	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.w(x:C-->(A X. B) /\ y = <.D, R>.))
18		ffvelrn 3814	. . . . . . . . 9 |- ((x:C-->(A X. B) /\ z e. C) -> (x` z) e. (A X. B))
19		elxp4 3453	. . . . . . . . . 10 |- ((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	pm3.26bi 322	. . . . . . . . 9 |- ((x` z) e. (A X. B) -> (x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
21	18, 20	syl 10	. . . . . . . 8 |- ((x:C-->(A X. B) /\ z e. C) -> (x` z) = <.U.dom {(x` z)}, U.ran {(x` z)}>.)
22	21	adantlr 393	. . . . . . 7 |- (((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 4497	. . . . . . . . 9 |- ((y = <.D, R>. /\ z e. C) -> ((U.dom { y}` z) = U.dom {(x` z)} /\ (U.ran { y}` z) = U.ran {(x` z)}))
24		opeq12 2489	. . . . . . . . 9 |- (((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 10	. . . . . . . 8 |- ((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 392	. . . . . . 7 |- (((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 1510	. . . . . 6 |- (((x:C-->(A X. B) /\ y = <.D, R>.) /\ z e. C) -> (x` z) = <.(U.dom { y}` z), (U.ran { y}` z)>.)
28	27	eqeq2d 1486	. . . . 5 |- (((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	pm5.32da 649	. . . 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)>.)))
30	14, 17, 29	opabbid 2669	. . 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)>.)})
31	30, 11	syl6eqr 1525	. 2 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> {<.z, w>. | (z e. C /\ w = (x` z))} = S)
32	4, 31	eqtrd 1507	1 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = S)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  <.cop 2411  U.cuni 2503  {copab 2666   X. cxp 3168  dom cdm 3170  ran crn 3171   Fn wfn 3177  -->wf 3178  ` cfv 3182

This theorem is referenced by:  xpmapenlem5 4500

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198

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