Theorem frd 5922

Description: Founded relationship in natural deduction form. (Contributed by SF, 12-Mar-2015.)

Hypotheses

Ref	Expression
frd.1	|- (ph -> R Fr A)
frd.2	|- (ph -> X e. V)
frd.3	|- (ph -> X C_ A)
frd.4	|- (ph -> X =/= (/))

Assertion

Ref	Expression
frd	|- (ph -> E.y e. X A.z e. X (zRy -> z = y))

Distinct variable groups:   y,R,z   y,X,z

Allowed substitution hints:   ph(y,z)   A(y,z)   V(y,z)

Proof of Theorem frd

Dummy variables a r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frd.3	. 2 |- (ph -> X C_ A)
2		frd.4	. 2 |- (ph -> X =/= (/))
3		frd.2	. . 3 |- (ph -> X e. V)
4		frd.1	. . . 4 |- (ph -> R Fr A)
5		brex 4689	. . . . . 6 |- (R Fr A -> (R e. _V /\ A e. _V))
6		breq 4641	. . . . . . . . . . 11 |- (r = R -> (zry <-> zRy))
7	6	imbi1d 308	. . . . . . . . . 10 |- (r = R -> ((zry -> z = y) <-> (zRy -> z = y)))
8	7	rexralbidv 2658	. . . . . . . . 9 |- (r = R -> (E.y e. x A.z e. x (zry -> z = y) <-> E.y e. x A.z e. x (zRy -> z = y)))
9	8	imbi2d 307	. . . . . . . 8 |- (r = R -> (((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zry -> z = y)) <-> ((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
10	9	albidv 1625	. . . . . . 7 |- (r = R -> (A.x((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zry -> z = y)) <-> A.x((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
11		sseq2 3293	. . . . . . . . . 10 |- (a = A -> (x C_ a <-> x C_ A))
12	11	anbi1d 685	. . . . . . . . 9 |- (a = A -> ((x C_ a /\ x =/= (/)) <-> (x C_ A /\ x =/= (/))))
13	12	imbi1d 308	. . . . . . . 8 |- (a = A -> (((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)) <-> ((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
14	13	albidv 1625	. . . . . . 7 |- (a = A -> (A.x((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)) <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
15		df-found 5905	. . . . . . 7 |- Fr = {<.r, a>. | A.x((x C_ a /\ x =/= (/)) -> E.y e. x A.z e. x (zry -> z = y))}
16	10, 14, 15	brabg 4706	. . . . . 6 |- ((R e. _V /\ A e. _V) -> (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
17	5, 16	syl 15	. . . . 5 |- (R Fr A -> (R Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y))))
18	17	ibi 232	. . . 4 |- (R Fr A -> A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)))
19	4, 18	syl 15	. . 3 |- (ph -> A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)))
20		sseq1 3292	. . . . . 6 |- (x = X -> (x C_ A <-> X C_ A))
21		neeq1 2524	. . . . . 6 |- (x = X -> (x =/= (/) <-> X =/= (/)))
22	20, 21	anbi12d 691	. . . . 5 |- (x = X -> ((x C_ A /\ x =/= (/)) <-> (X C_ A /\ X =/= (/))))
23		raleq 2807	. . . . . 6 |- (x = X -> (A.z e. x (zRy -> z = y) <-> A.z e. X (zRy -> z = y)))
24	23	rexeqbi1dv 2816	. . . . 5 |- (x = X -> (E.y e. x A.z e. x (zRy -> z = y) <-> E.y e. X A.z e. X (zRy -> z = y)))
25	22, 24	imbi12d 311	. . . 4 |- (x = X -> (((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)) <-> ((X C_ A /\ X =/= (/)) -> E.y e. X A.z e. X (zRy -> z = y))))
26	25	spcgv 2939	. . 3 |- (X e. V -> (A.x((x C_ A /\ x =/= (/)) -> E.y e. x A.z e. x (zRy -> z = y)) -> ((X C_ A /\ X =/= (/)) -> E.y e. X A.z e. X (zRy -> z = y))))
27	3, 19, 26	sylc 56	. 2 |- (ph -> ((X C_ A /\ X =/= (/)) -> E.y e. X A.z e. X (zRy -> z = y)))
28	1, 2, 27	mp2and 660	1 |- (ph -> E.y e. X A.z e. X (zRy -> z = y))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   =/= wne 2516  A.wral 2614  E.wrex 2615  _Vcvv 2859   C_ wss 3257  (/)c0 3550   class class class wbr 4639   Fr cfound 5894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-found 5905

This theorem is referenced by:  frds  5935