Theorem xpmapenlem3 4487

Description: Lemma for xpmapen 4490.

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 3623	. . . 4 |- (x:C-->(A X. B) -> x Fn C)
2		fnopabfv 3753	. . . 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-17 970	. . . . 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 4485	. . . . . 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 1008	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.z(x:C-->(A X. B) /\ y = <.D, R>.))
15		ax-17 970	. . . . 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 1008	. . . 4 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> A.w(x:C-->(A X. B) /\ y = <.D, R>.))
18		ffvelrn 3809	. . . . . . . . 9 |- ((x:C-->(A X. B) /\ z e. C) -> (x` z) e. (A X. B))
19		elxp4 3449	. . . . . . . . . 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 4486	. . . . . . . . 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 2486	. . . . . . . . 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 1508	. . . . . 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 1484	. . . . 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 648	. . . 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 2665	. . 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 1523	. 2 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> {<.z, w>. | (z e. C /\ w = (x` z))} = S)
32	4, 31	eqtrd 1505	1 |- ((x:C-->(A X. B) /\ y = <.D, R>.) -> x = S)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  Vcvv 1808  {csn 2406  <.cop 2408  U.cuni 2499  {copab 2662   X. cxp 3164  dom cdm 3166  ran crn 3167   Fn wfn 3173  -->wf 3174  ` cfv 3178

This theorem is referenced by:  xpmapenlem5 4489

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194

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