Theorem elrnopabg 3800

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 1817	. . 3 |- (B e. D -> B e. V)
2	1	r19.20si 1706	. 2 |- (A.x e. A B e. D -> A.x e. A B e. V)
3		eueq 1916	. . . . . 6 |- (B e. V <-> E!y y = B)
4	3	biimp 151	. . . . 5 |- (B e. V -> E!y y = B)
5	4	r19.20si 1706	. . . 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 3615	. . . 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 3760	. . . 4 |- (F Fn A -> (C e. ran F <-> E.z e. A (F` z) = C))
10		hbra1 1687	. . . . . 6 |- (A.x e. A B e. V -> A.xA.x e. A B e. V)
11		ra4 1694	. . . . . . . . 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 3791	. . . . . . . . . 10 |- ((x e. A /\ B e. V) -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
15	6	fveq1i 3725	. . . . . . . . . 10 |- (F` x) = ({<.x, y>. | (x e. A /\ y = B)}` x)
16	14, 15	syl5eq 1519	. . . . . . . . 9 |- ((x e. A /\ B e. V) -> (F` x) = B)
17	16	eqeq1d 1483	. . . . . . . 8 |- ((x e. A /\ B e. V) -> ((F` x) = C <-> B = C))
18		eqcom 1477	. . . . . . . 8 |- (B = C <-> C = B)
19	17, 18	syl6bb 536	. . . . . . 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 1658	. . . . 5 |- (A.x e. A B e. V -> (E.x e. A (F` x) = C <-> E.x e. A C = B))
22		hbopab1 2813	. . . . . . . . 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 1563	. . . . . . . 8 |- (w e. F -> A.x w e. F)
24		ax-17 971	. . . . . . . 8 |- (w e. z -> A.x w e. z)
25	23, 24	hbfv 3729	. . . . . . 7 |- (w e. (F` z) -> A.x w e. (F` z))
26		ax-17 971	. . . . . . 7 |- (w e. C -> A.x w e. C)
27	25, 26	hbeq 1565	. . . . . 6 |- ((F` z) = C -> A.x(F` z) = C)
28		ax-17 971	. . . . . 6 |- ((F` x) = C -> A.z(F` x) = C)
29		fveq2 3724	. . . . . . 7 |- (z = x -> (F` z) = (F` x))
30	29	eqeq1d 1483	. . . . . 6 |- (z = x -> ((F` z) = C <-> (F` x) = C))
31	27, 28, 30	cbvrex 1799	. . . . 5 |- (E.z e. A (F` z) = C <-> E.x e. A (F` x) = C)
32	21, 31	syl5bb 532	. . . 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 541	. . 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 704	. 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 956   e. wcel 958  E!weu 1380  A.wral 1645  E.wrex 1646  Vcvv 1811  {copab 2666  ran crn 3171   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  elrnopab 3801

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-sep 2703  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-fv 3198

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