Theorem onopruni 5117

Description: A variant of onfununi 5116 for operations. (Contributed by Eric Schmidt, 26-May-2009.)

Hypotheses

Ref	Expression
onopruni.1	|- (Lim y -> (AFy) = U_x e. y (AFx))
onopruni.2	|- ((x e. On /\ y e. On /\ x C_ y) -> (AFx) C_ (AFy))

Assertion

Ref	Expression
onopruni	|- ((S e. T /\ S C_ On /\ S =/= (/)) -> (AFU.S) = U_x e. S (AFx))

Distinct variable groups:   x,y,A   x,F,y   x,S,y   x,T

Proof of Theorem onopruni

Step	Hyp	Ref	Expression
1		onopruni.1	. . . 4 |- (Lim y -> (AFy) = U_x e. y (AFx))
2		visset 2295	. . . . 5 |- y e. _V
3		oprex 4907	. . . . 5 |- (AFy) e. _V
4		opreq2 4890	. . . . 5 |- (z = y -> (AFz) = (AFy))
5	2, 3, 4	fvopab 4753	. . . 4 |- ({<.z, w>. | w = (AFz)}` y) = (AFy)
6		visset 2295	. . . . . . 7 |- x e. _V
7		oprex 4907	. . . . . . 7 |- (AFx) e. _V
8		opreq2 4890	. . . . . . 7 |- (z = x -> (AFz) = (AFx))
9	6, 7, 8	fvopab 4753	. . . . . 6 |- ({<.z, w>. | w = (AFz)}` x) = (AFx)
10	9	a1i 8	. . . . 5 |- (x e. y -> ({<.z, w>. | w = (AFz)}` x) = (AFx))
11	10	iuneq2i 3276	. . . 4 |- U_x e. y ({<.z, w>. | w = (AFz)}` x) = U_x e. y (AFx)
12	1, 5, 11	3eqtr4g 1953	. . 3 |- (Lim y -> ({<.z, w>. | w = (AFz)}` y) = U_x e. y ({<.z, w>. | w = (AFz)}` x))
13		onopruni.2	. . . 4 |- ((x e. On /\ y e. On /\ x C_ y) -> (AFx) C_ (AFy))
14	13, 9, 5	3sstr4g 2658	. . 3 |- ((x e. On /\ y e. On /\ x C_ y) -> ({<.z, w>. | w = (AFz)}` x) C_ ({<.z, w>. | w = (AFz)}` y))
15	12, 14	onfununi 5116	. 2 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> ({<.z, w>. | w = (AFz)}` U.S) = U_x e. S ({<.z, w>. | w = (AFz)}` x))
16		opreq2 4890	. . . . 5 |- (z = U.S -> (AFz) = (AFU.S))
17	16	fvopabg 4748	. . . 4 |- ((U.S e. _V /\ (AFU.S) e. _V) -> ({<.z, w>. | w = (AFz)}` U.S) = (AFU.S))
18		uniexg 3795	. . . 4 |- (S e. T -> U.S e. _V)
19		oprex 4907	. . . 4 |- (AFU.S) e. _V
20	17, 18, 19	sylancl 525	. . 3 |- (S e. T -> ({<.z, w>. | w = (AFz)}` U.S) = (AFU.S))
21	20	3ad2ant1 897	. 2 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> ({<.z, w>. | w = (AFz)}` U.S) = (AFU.S))
22	9	a1i 8	. . . 4 |- (x e. S -> ({<.z, w>. | w = (AFz)}` x) = (AFx))
23	22	iuneq2i 3276	. . 3 |- U_x e. S ({<.z, w>. | w = (AFz)}` x) = U_x e. S (AFx)
24	23	a1i 8	. 2 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> U_x e. S ({<.z, w>. | w = (AFz)}` x) = U_x e. S (AFx))
25	15, 21, 24	3eqtr3d 1934	1 |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (AFU.S) = U_x e. S (AFx))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U.cuni 3177  U_ciun 3255  {copab 3395  Oncon0 3657  Lim wlim 3658  ` cfv 3998  (class class class)co 4884

This theorem is referenced by:  onopriun 5118

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886

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