Theorem rankuni 9552

Description: The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankuni	⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)

Proof of Theorem rankuni

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4847	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	fveq2d 6760	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) = (rank‘∪ 𝐴))
3		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	unieqd 4850	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ (rank‘𝑥) = ∪ (rank‘𝐴))
5	2, 4	eqeq12d 2754	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) = ∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)))
6		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankuni2 9544	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧)
8		fvex 6769	. . . . . . 7 ⊢ (rank‘𝑧) ∈ V
9	8	dfiun2 4959	. . . . . 6 ⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
10	7, 9	eqtri 2766	. . . . 5 ⊢ (rank‘∪ 𝑥) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
11		df-rex 3069	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)))
12	6	rankel 9528	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
13	12	anim1i 614	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
14	13	eximi 1838	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15		19.42v 1958	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
17	16	pm5.32ri 575	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
18	17	exbii 1851	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20		rankon 9484	. . . . . . . . . . . . . . . . 17 ⊢ (rank‘𝑥) ∈ On
21	20	oneli 6359	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22		r1fnon 9456	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
23		fndm 6520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
25	21, 24	eleqtrrdi 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26		rankr1id 9551	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝑦)) = 𝑦)
27	25, 26	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1‘𝑦)) = 𝑦)
28	27	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1‘𝑦)))
29		fvex 6769	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝑦) ∈ V
30		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑦) → (rank‘𝑧) = (rank‘(𝑅1‘𝑦)))
31	30	eqeq2d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1‘𝑦))))
32	29, 31	spcev 3535	. . . . . . . . . . . . 13 ⊢ (𝑦 = (rank‘(𝑅1‘𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
33	28, 32	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
34	33	ancli 548	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
35	19, 34	impbii 208	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
36	15, 18, 35	3bitr3i 300	. . . . . . . . 9 ⊢ (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
37	14, 36	sylib 217	. . . . . . . 8 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
38	11, 37	sylbi 216	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
39	38	abssi 3999	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
40	39	unissi 4845	. . . . 5 ⊢ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪ (rank‘𝑥)
41	10, 40	eqsstri 3951	. . . 4 ⊢ (rank‘∪ 𝑥) ⊆ ∪ (rank‘𝑥)
42		pwuni 4875	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥
43		vuniex 7570	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
44	43	pwex 5298	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝑥 ∈ V
45	44	rankss 9538	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥))
46	42, 45	ax-mp 5	. . . . . . 7 ⊢ (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥)
47	43	rankpw 9532	. . . . . . 7 ⊢ (rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥)
48	46, 47	sseqtri 3953	. . . . . 6 ⊢ (rank‘𝑥) ⊆ suc (rank‘∪ 𝑥)
49	48	unissi 4845	. . . . 5 ⊢ ∪ (rank‘𝑥) ⊆ ∪ suc (rank‘∪ 𝑥)
50		rankon 9484	. . . . . 6 ⊢ (rank‘∪ 𝑥) ∈ On
51	50	onunisuci 6365	. . . . 5 ⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥)
52	49, 51	sseqtri 3953	. . . 4 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥)
53	41, 52	eqssi 3933	. . 3 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
54	5, 53	vtoclg 3495	. 2 ⊢ (𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
55		uniexb 7592	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
56		fvprc 6748	. . . . 5 ⊢ (¬ ∪ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
57	55, 56	sylnbi 329	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
58		uni0 4866	. . . 4 ⊢ ∪ ∅ = ∅
59	57, 58	eqtr4di 2797	. . 3 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ ∅)
60		fvprc 6748	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
61	60	unieqd 4850	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ (rank‘𝐴) = ∪ ∅)
62	59, 61	eqtr4d 2781	. 2 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
63	54, 62	pm2.61i 182	1 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ∪ ciun 4921  dom cdm 5580  Oncon0 6251  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  𝑅₁cr1 9451  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  rankuniss  9555  rankbnd2  9558  rankxplim2  9569  rankxplim3  9570  rankxpsuc  9571  r1limwun  10423  hfuni  34413