Theorem rankuni 9837

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 4887	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	fveq2d 6888	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) = (rank‘∪ 𝐴))
3		fveq2 6884	. . . . 5 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	unieqd 4889	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ (rank‘𝑥) = ∪ (rank‘𝐴))
5	2, 4	eqeq12d 2785	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) = ∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)))
6		vex 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankuni2 9829	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧)
8		fvex 6897	. . . . . . 7 ⊢ (rank‘𝑧) ∈ V
9	8	dfiun2 5000	. . . . . 6 ⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
10	7, 9	eqtri 2792	. . . . 5 ⊢ (rank‘∪ 𝑥) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
11		df-rex 3096	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)))
12	6	rankel 9813	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
13	12	anim1i 626	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
14	13	eximi 1862	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15		19.42v 1980	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16		eleq1 2857	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
17	16	pm5.32ri 585	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
18	17	exbii 1875	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19		simpl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20		rankon 9769	. . . . . . . . . . . . . . . . 17 ⊢ (rank‘𝑥) ∈ On
21	20	oneli 6479	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22		r1fnon 9741	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
23		fndm 6641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
25	21, 24	eleqtrrdi 2880	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26		rankr1id 9836	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝑦)) = 𝑦)
27	25, 26	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1‘𝑦)) = 𝑦)
28	27	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1‘𝑦)))
29		fvex 6897	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝑦) ∈ V
30		fveq2 6884	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑦) → (rank‘𝑧) = (rank‘(𝑅1‘𝑦)))
31	30	eqeq2d 2780	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1‘𝑦))))
32	29, 31	spcev 3574	. . . . . . . . . . . . 13 ⊢ (𝑦 = (rank‘(𝑅1‘𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
33	28, 32	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
34	33	ancli 557	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
35	19, 34	impbii 212	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
36	15, 18, 35	3bitr3i 304	. . . . . . . . 9 ⊢ (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
37	14, 36	sylib 221	. . . . . . . 8 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
38	11, 37	sylbi 220	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
39	38	abssi 4030	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
40	39	unissi 4885	. . . . 5 ⊢ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪ (rank‘𝑥)
41	10, 40	eqsstri 3991	. . . 4 ⊢ (rank‘∪ 𝑥) ⊆ ∪ (rank‘𝑥)
42		pwuni 4915	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥
43		vuniex 7740	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
44	43	pwex 5354	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝑥 ∈ V
45	44	rankss 9823	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥))
46	42, 45	ax-mp 5	. . . . . . 7 ⊢ (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥)
47	43	rankpw 9817	. . . . . . 7 ⊢ (rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥)
48	46, 47	sseqtri 3993	. . . . . 6 ⊢ (rank‘𝑥) ⊆ suc (rank‘∪ 𝑥)
49	48	unissi 4885	. . . . 5 ⊢ ∪ (rank‘𝑥) ⊆ ∪ suc (rank‘∪ 𝑥)
50		rankon 9769	. . . . . 6 ⊢ (rank‘∪ 𝑥) ∈ On
51	50	onunisuci 6485	. . . . 5 ⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥)
52	49, 51	sseqtri 3993	. . . 4 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥)
53	41, 52	eqssi 3961	. . 3 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
54	5, 53	vtoclg 3531	. 2 ⊢ (𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
55		uniexb 7765	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
56		fvprc 6876	. . . . 5 ⊢ (¬ ∪ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
57	55, 56	sylnbi 333	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
58		uni0 4905	. . . 4 ⊢ ∪ ∅ = ∅
59	57, 58	eqtr4di 2822	. . 3 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ ∅)
60		fvprc 6876	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
61	60	unieqd 4889	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ (rank‘𝐴) = ∪ ∅)
62	59, 61	eqtr4d 2807	. 2 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
63	54, 62	pm2.61i 184	1 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  ∪ ciun 4960  dom cdm 5664  Oncon0 6363  suc csuc 6365   Fn wfn 6534  ‘cfv 6539  𝑅₁cr1 9736  rankcrnk 9737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-reg 9556  ax-inf2 9612

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-r1 9738  df-rank 9739

This theorem is referenced by:  rankuniss  9840  rankbnd2  9843  rankxplim2  9854  rankxplim3  9855  rankxpsuc  9856  r1limwun  10723  hfuni  36611