Theorem rankuni 5809

Description: The rank of a union. Part of Exercise 4 of [Kunen] p. 107.

Assertion

Ref	Expression
rankuni	|- (rank` U.A) = U.(rank` A)

Proof of Theorem rankuni

Step	Hyp	Ref	Expression
1		unieq 3185	. . . . 5 |- (x = A -> U.x = U.A)
2	1	fveq2d 4685	. . . 4 |- (x = A -> (rank` U.x) = (rank` U.A))
3		fveq2 4681	. . . . 5 |- (x = A -> (rank` x) = (rank` A))
4	3	unieqd 3188	. . . 4 |- (x = A -> U.(rank` x) = U.(rank` A))
5	2, 4	eqeq12d 1899	. . 3 |- (x = A -> ((rank` U.x) = U.(rank` x) <-> (rank` U.A) = U.(rank` A)))
6		visset 2295	. . . . . . 7 |- x e. _V
7	6	rankuni2 5801	. . . . . 6 |- (rank` U.x) = U_z e. x (rank` z)
8		fvex 4689	. . . . . . 7 |- (rank` z) e. _V
9	8	dfiun2 3285	. . . . . 6 |- U_z e. x (rank` z) = U.{y | E.z e. x y = (rank` z)}
10	7, 9	eqtri 1908	. . . . 5 |- (rank` U.x) = U.{y | E.z e. x y = (rank` z)}
11		df-rex 2110	. . . . . . . 8 |- (E.z e. x y = (rank` z) <-> E.z(z e. x /\ y = (rank` z)))
12	6	rankel 5791	. . . . . . . . . . 11 |- (z e. x -> (rank` z) e. (rank` x))
13	12	anim1i 361	. . . . . . . . . 10 |- ((z e. x /\ y = (rank` z)) -> ((rank` z) e. (rank` x) /\ y = (rank` z)))
14	13	eximi 1387	. . . . . . . . 9 |- (E.z(z e. x /\ y = (rank` z)) -> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
15		19.42v 1688	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> (y e. (rank` x) /\ E.z y = (rank` z)))
16		eleq1 1957	. . . . . . . . . . . 12 |- (y = (rank` z) -> (y e. (rank` x) <-> (rank` z) e. (rank` x)))
17	16	pm5.32ri 708	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ y = (rank` z)) <-> ((rank` z) e. (rank` x) /\ y = (rank` z)))
18	17	exbii 1398	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
19		simpl 346	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ E.z y = (rank` z)) -> y e. (rank` x))
20		rankon 5782	. . . . . . . . . . . . . . . 16 |- (rank` x) e. On
21	20	oneli 3777	. . . . . . . . . . . . . . 15 |- (y e. (rank` x) -> y e. On)
22		rankr1id 5808	. . . . . . . . . . . . . . 15 |- (y e. On <-> (rank` (R1` y)) = y)
23	21, 22	sylib 215	. . . . . . . . . . . . . 14 |- (y e. (rank` x) -> (rank` (R1` y)) = y)
24	23	eqcomd 1889	. . . . . . . . . . . . 13 |- (y e. (rank` x) -> y = (rank` (R1` y)))
25		fvex 4689	. . . . . . . . . . . . . 14 |- (R1` y) e. _V
26		fveq2 4681	. . . . . . . . . . . . . . 15 |- (z = (R1` y) -> (rank` z) = (rank` (R1` y)))
27	26	eqeq2d 1895	. . . . . . . . . . . . . 14 |- (z = (R1` y) -> (y = (rank` z) <-> y = (rank` (R1` y))))
28	25, 27	cla4ev 2371	. . . . . . . . . . . . 13 |- (y = (rank` (R1` y)) -> E.z y = (rank` z))
29	24, 28	syl 12	. . . . . . . . . . . 12 |- (y e. (rank` x) -> E.z y = (rank` z))
30	29	ancli 320	. . . . . . . . . . 11 |- (y e. (rank` x) -> (y e. (rank` x) /\ E.z y = (rank` z)))
31	19, 30	impbii 174	. . . . . . . . . 10 |- ((y e. (rank` x) /\ E.z y = (rank` z)) <-> y e. (rank` x))
32	15, 18, 31	3bitr3i 198	. . . . . . . . 9 |- (E.z((rank` z) e. (rank` x) /\ y = (rank` z)) <-> y e. (rank` x))
33	14, 32	sylib 215	. . . . . . . 8 |- (E.z(z e. x /\ y = (rank` z)) -> y e. (rank` x))
34	11, 33	sylbi 216	. . . . . . 7 |- (E.z e. x y = (rank` z) -> y e. (rank` x))
35	34	abssi 2682	. . . . . 6 |- {y | E.z e. x y = (rank` z)} C_ (rank` x)
36		uniss 3199	. . . . . 6 |- ({y | E.z e. x y = (rank` z)} C_ (rank` x) -> U.{y | E.z e. x y = (rank` z)} C_ U.(rank` x))
37	35, 36	ax-mp 7	. . . . 5 |- U.{y | E.z e. x y = (rank` z)} C_ U.(rank` x)
38	10, 37	eqsstri 2647	. . . 4 |- (rank` U.x) C_ U.(rank` x)
39		pwuni 3505	. . . . . . . 8 |- x C_ ~PU.x
40	6	uniex 3794	. . . . . . . . . 10 |- U.x e. _V
41	40	pwex 3487	. . . . . . . . 9 |- ~PU.x e. _V
42	41	rankss 5799	. . . . . . . 8 |- (x C_ ~PU.x -> (rank` x) C_ (rank` ~PU.x))
43	39, 42	ax-mp 7	. . . . . . 7 |- (rank` x) C_ (rank` ~PU.x)
44	40	rankpw 5795	. . . . . . 7 |- (rank` ~PU.x) = suc (rank` U.x)
45	43, 44	sseqtri 2649	. . . . . 6 |- (rank` x) C_ suc (rank` U.x)
46		uniss 3199	. . . . . 6 |- ((rank` x) C_ suc (rank` U.x) -> U.(rank` x) C_ U.suc (rank` U.x))
47	45, 46	ax-mp 7	. . . . 5 |- U.(rank` x) C_ U.suc (rank` U.x)
48		rankon 5782	. . . . . 6 |- (rank` U.x) e. On
49	48	onunisuci 3783	. . . . 5 |- U.suc (rank` U.x) = (rank` U.x)
50	47, 49	sseqtri 2649	. . . 4 |- U.(rank` x) C_ (rank` U.x)
51	38, 50	eqssi 2632	. . 3 |- (rank` U.x) = U.(rank` x)
52	5, 51	vtoclg 2346	. 2 |- (A e. _V -> (rank` U.A) = U.(rank` A))
53		uniexb 3851	. . . . . 6 |- (A e. _V <-> U.A e. _V)
54	53	notbii 204	. . . . 5 |- (-. A e. _V <-> -. U.A e. _V)
55		fvprc 4678	. . . . 5 |- (-. U.A e. _V -> (rank` U.A) = (/))
56	54, 55	sylbi 216	. . . 4 |- (-. A e. _V -> (rank` U.A) = (/))
57		uni0 3205	. . . 4 |- U.(/) = (/)
58	56, 57	syl6eqr 1946	. . 3 |- (-. A e. _V -> (rank` U.A) = U.(/))
59		fvprc 4678	. . . 4 |- (-. A e. _V -> (rank` A) = (/))
60	59	unieqd 3188	. . 3 |- (-. A e. _V -> U.(rank` A) = U.(/))
61	58, 60	eqtr4d 1928	. 2 |- (-. A e. _V -> (rank` U.A) = U.(rank` A))
62	52, 61	pm2.61i 140	1 |- (rank` U.A) = U.(rank` A)

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177  U_ciun 3255  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankuniss 5812  rankbnd2 5815  rankxplim2 5824  rankxplim3 5825  rankxpsuc 5826

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

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-rab 2112  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-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  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-suc 3663  df-om 3950  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-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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