Theorem elrnopabg 3795

Description: Membership in the range of an ordered pair class abstraction.

Hypothesis

Ref	Expression
elrnopabg.1	|- F = {<.x, y>. | (x e. A /\ y = B)}

Assertion

Ref	Expression
elrnopabg	|- (A.x e. A B e. D -> (C e. ran F <-> E.x e. A C = B))

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

Proof of Theorem elrnopabg

Step	Hyp	Ref	Expression
1		elisset 1814	. . 3 |- (B e. D -> B e. V)
2	1	r19.20si 1704	. 2 |- (A.x e. A B e. D -> A.x e. A B e. V)
3		eueq 1913	. . . . . 6 |- (B e. V <-> E!y y = B)
4	3	biimp 151	. . . . 5 |- (B e. V -> E!y y = B)
5	4	r19.20si 1704	. . . 4 |- (A.x e. A B e. V -> A.x e. A E!y y = B)
6		elrnopabg.1	. . . . 5 |- F = {<.x, y>. | (x e. A /\ y = B)}
7	6	fnopabg 3611	. . . 4 |- (A.x e. A E!y y = B <-> F Fn A)
8	5, 7	sylib 198	. . 3 |- (A.x e. A B e. V -> F Fn A)
9		fvelrnb 3755	. . . 4 |- (F Fn A -> (C e. ran F <-> E.z e. A (F` z) = C))
10		hbra1 1685	. . . . . 6 |- (A.x e. A B e. V -> A.xA.x e. A B e. V)
11		ra4 1692	. . . . . . . . 9 |- (A.x e. A B e. V -> (x e. A -> B e. V))
12	11	ancld 298	. . . . . . . 8 |- (A.x e. A B e. V -> (x e. A -> (x e. A /\ B e. V)))
13	12	imp 350	. . . . . . 7 |- ((A.x e. A B e. V /\ x e. A) -> (x e. A /\ B e. V))
14		fvopab2 3786	. . . . . . . . . 10 |- ((x e. A /\ B e. V) -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
15	6	fveq1i 3720	. . . . . . . . . 10 |- (F` x) = ({<.x, y>. | (x e. A /\ y = B)}` x)
16	14, 15	syl5eq 1517	. . . . . . . . 9 |- ((x e. A /\ B e. V) -> (F` x) = B)
17	16	eqeq1d 1481	. . . . . . . 8 |- ((x e. A /\ B e. V) -> ((F` x) = C <-> B = C))
18		eqcom 1475	. . . . . . . 8 |- (B = C <-> C = B)
19	17, 18	syl6bb 535	. . . . . . 7 |- ((x e. A /\ B e. V) -> ((F` x) = C <-> C = B))
20	13, 19	syl 10	. . . . . 6 |- ((A.x e. A B e. V /\ x e. A) -> ((F` x) = C <-> C = B))
21	10, 20	rexbida 1656	. . . . 5 |- (A.x e. A B e. V -> (E.x e. A (F` x) = C <-> E.x e. A C = B))
22		hbopab1 2809	. . . . . . . . 9 |- (w e. {<.x, y>. | (x e. A /\ y = B)} -> A.x w e. {<.x, y>. | (x e. A /\ y = B)})
23	6, 22	hbxfr 1561	. . . . . . . 8 |- (w e. F -> A.x w e. F)
24		ax-17 970	. . . . . . . 8 |- (w e. z -> A.x w e. z)
25	23, 24	hbfv 3724	. . . . . . 7 |- (w e. (F` z) -> A.x w e. (F` z))
26		ax-17 970	. . . . . . 7 |- (w e. C -> A.x w e. C)
27	25, 26	hbeq 1563	. . . . . 6 |- ((F` z) = C -> A.x(F` z) = C)
28		ax-17 970	. . . . . 6 |- ((F` x) = C -> A.z(F` x) = C)
29		fveq2 3719	. . . . . . 7 |- (z = x -> (F` z) = (F` x))
30	29	eqeq1d 1481	. . . . . 6 |- (z = x -> ((F` z) = C <-> (F` x) = C))
31	27, 28, 30	cbvrex 1796	. . . . 5 |- (E.z e. A (F` z) = C <-> E.x e. A (F` x) = C)
32	21, 31	syl5bb 531	. . . 4 |- (A.x e. A B e. V -> (E.z e. A (F` z) = C <-> E.x e. A C = B))
33	9, 32	sylan9bbr 540	. . 3 |- ((A.x e. A B e. V /\ F Fn A) -> (C e. ran F <-> E.x e. A C = B))
34	8, 33	mpdan 703	. 2 |- (A.x e. A B e. V -> (C e. ran F <-> E.x e. A C = B))
35	2, 34	syl 10	1 |- (A.x e. A B e. D -> (C e. ran F <-> E.x e. A C = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  E!weu 1379  A.wral 1643  E.wrex 1644  Vcvv 1808  {copab 2662  ran crn 3167   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  elrnopab 3796

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-sep 2699  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-fv 3194

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