Theorem rankuni 4708

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 2514	. . . . 5 |- (x = A -> U.x = U.A)
2	1	fveq2d 3734	. . . 4 |- (x = A -> (rank` U.x) = (rank` U.A))
3		fveq2 3730	. . . . 5 |- (x = A -> (rank` x) = (rank` A))
4	3	unieqd 2516	. . . 4 |- (x = A -> U.(rank` x) = U.(rank` A))
5	2, 4	eqeq12d 1492	. . 3 |- (x = A -> ((rank` U.x) = U.(rank` x) <-> (rank` U.A) = U.(rank` A)))
6		visset 1816	. . . . . . 7 |- x e. V
7	6	rankuni2 4700	. . . . . 6 |- (rank` U.x) = U_z e. x (rank` z)
8		fvex 3738	. . . . . . 7 |- (rank` z) e. V
9	8	dfiun2 2591	. . . . . 6 |- U_z e. x (rank` z) = U.{y | E.z e. x y = (rank` z)}
10	7, 9	eqtr 1498	. . . . 5 |- (rank` U.x) = U.{y | E.z e. x y = (rank` z)}
11		df-rex 1653	. . . . . . . 8 |- (E.z e. x y = (rank` z) <-> E.z(z e. x /\ y = (rank` z)))
12	6	rankel 4690	. . . . . . . . . . 11 |- (z e. x -> (rank` z) e. (rank` x))
13	12	anim1i 334	. . . . . . . . . 10 |- ((z e. x /\ y = (rank` z)) -> ((rank` z) e. (rank` x) /\ y = (rank` z)))
14	13	19.22i 1042	. . . . . . . . 9 |- (E.z(z e. x /\ y = (rank` z)) -> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
15		19.42v 1310	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> (y e. (rank` x) /\ E.z y = (rank` z)))
16		eleq1 1537	. . . . . . . . . . . 12 |- (y = (rank` z) -> (y e. (rank` x) <-> (rank` z) e. (rank` x)))
17	16	pm5.32ri 648	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ y = (rank` z)) <-> ((rank` z) e. (rank` x) /\ y = (rank` z)))
18	17	exbii 1053	. . . . . . . . . 10 |- (E.z(y e. (rank` x) /\ y = (rank` z)) <-> E.z((rank` z) e. (rank` x) /\ y = (rank` z)))
19		pm3.26 319	. . . . . . . . . . 11 |- ((y e. (rank` x) /\ E.z y = (rank` z)) -> y e. (rank` x))
20		rankon 4681	. . . . . . . . . . . . . . . 16 |- (rank` x) e. On
21	20	onel 3104	. . . . . . . . . . . . . . 15 |- (y e. (rank` x) -> y e. On)
22		rankr1id 4707	. . . . . . . . . . . . . . 15 |- (y e. On <-> (rank` (R1` y)) = y)
23	21, 22	sylib 198	. . . . . . . . . . . . . 14 |- (y e. (rank` x) -> (rank` (R1` y)) = y)
24	23	eqcomd 1483	. . . . . . . . . . . . 13 |- (y e. (rank` x) -> y = (rank` (R1` y)))
25		fvex 3738	. . . . . . . . . . . . . 14 |- (R1` y) e. V
26		fveq2 3730	. . . . . . . . . . . . . . 15 |- (z = (R1` y) -> (rank` z) = (rank` (R1` y)))
27	26	eqeq2d 1489	. . . . . . . . . . . . . 14 |- (z = (R1` y) -> (y = (rank` z) <-> y = (rank` (R1` y))))
28	25, 27	cla4ev 1872	. . . . . . . . . . . . 13 |- (y = (rank` (R1` y)) -> E.z y = (rank` z))
29	24, 28	syl 10	. . . . . . . . . . . 12 |- (y e. (rank` x) -> E.z y = (rank` z))
30	29	ancli 296	. . . . . . . . . . 11 |- (y e. (rank` x) -> (y e. (rank` x) /\ E.z y = (rank` z)))
31	19, 30	impbi 157	. . . . . . . . . 10 |- ((y e. (rank` x) /\ E.z y = (rank` z)) <-> y e. (rank` x))
32	15, 18, 31	3bitr3 181	. . . . . . . . 9 |- (E.z((rank` z) e. (rank` x) /\ y = (rank` z)) <-> y e. (rank` x))
33	14, 32	sylib 198	. . . . . . . 8 |- (E.z(z e. x /\ y = (rank` z)) -> y e. (rank` x))
34	11, 33	sylbi 199	. . . . . . 7 |- (E.z e. x y = (rank` z) -> y e. (rank` x))
35	34	abssi 2125	. . . . . 6 |- {y | E.z e. x y = (rank` z)} (_ (rank` x)
36		uniss 2525	. . . . . 6 |- ({y | E.z e. x y = (rank` z)} (_ (rank` x) -> U.{y | E.z e. x y = (rank` z)} (_ U.(rank` x))
37	35, 36	ax-mp 7	. . . . 5 |- U.{y | E.z e. x y = (rank` z)} (_ U.(rank` x)
38	10, 37	eqsstr 2094	. . . 4 |- (rank` U.x) (_ U.(rank` x)
39		pwuni 2763	. . . . . . . 8 |- x (_ P~U.x
40	6	uniex 2876	. . . . . . . . . 10 |- U.x e. V
41	40	pwex 2751	. . . . . . . . 9 |- P~U.x e. V
42	41	rankss 4698	. . . . . . . 8 |- (x (_ P~U.x -> (rank` x) (_ (rank` P~U.x))
43	39, 42	ax-mp 7	. . . . . . 7 |- (rank` x) (_ (rank` P~U.x)
44	40	rankpw 4694	. . . . . . 7 |- (rank` P~U.x) = suc (rank` U.x)
45	43, 44	sseqtr 2096	. . . . . 6 |- (rank` x) (_ suc (rank` U.x)
46		uniss 2525	. . . . . 6 |- ((rank` x) (_ suc (rank` U.x) -> U.(rank` x) (_ U.suc (rank` U.x))
47	45, 46	ax-mp 7	. . . . 5 |- U.(rank` x) (_ U.suc (rank` U.x)
48		rankon 4681	. . . . . 6 |- (rank` U.x) e. On
49	48	onunisuc 3112	. . . . 5 |- U.suc (rank` U.x) = (rank` U.x)
50	47, 49	sseqtr 2096	. . . 4 |- U.(rank` x) (_ (rank` U.x)
51	38, 50	eqssi 2081	. . 3 |- (rank` U.x) = U.(rank` x)
52	5, 51	vtoclg 1850	. 2 |- (A e. V -> (rank` U.A) = U.(rank` A))
53		uniexb 2913	. . . . . 6 |- (A e. V <-> U.A e. V)
54	53	negbii 187	. . . . 5 |- (-. A e. V <-> -. U.A e. V)
55		fvprc 3727	. . . . 5 |- (-. U.A e. V -> (rank` U.A) = (/))
56	54, 55	sylbi 199	. . . 4 |- (-. A e. V -> (rank` U.A) = (/))
57		uni0 2529	. . . 4 |- U.(/) = (/)
58	56, 57	syl6eqr 1528	. . 3 |- (-. A e. V -> (rank` U.A) = U.(/))
59		fvprc 3727	. . . 4 |- (-. A e. V -> (rank` A) = (/))
60	59	unieqd 2516	. . 3 |- (-. A e. V -> U.(rank` A) = U.(/))
61	58, 60	eqtr4d 1513	. 2 |- (-. A e. V -> (rank` U.A) = U.(rank` A))
62	52, 61	pm2.61i 126	1 |- (rank` U.A) = U.(rank` A)

Syntax hints:  -. wn 2   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  U.cuni 2507  U_ciun 2570  Oncon0 2954  suc csuc 2956  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankuniss 4711  rankbnd2 4714  rankxplim2 4723  rankxplim3 4724  rankxpsuc 4725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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