Theorem rankuni 9856

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 4878	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	fveq2d 6885	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) = (rank‘∪ 𝐴))
3		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	unieqd 4880	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ (rank‘𝑥) = ∪ (rank‘𝐴))
5	2, 4	eqeq12d 2776	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) = ∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)))
6		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankuni2 9846	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧)
8		fvex 6894	. . . . . . 7 ⊢ (rank‘𝑧) ∈ V
9	8	dfiun2 4990	. . . . . 6 ⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
10	7, 9	eqtri 2783	. . . . 5 ⊢ (rank‘∪ 𝑥) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
11		df-rex 3087	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)))
12	6	rankel 9828	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
13	12	anim1i 627	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
14	13	eximi 1868	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15		19.42v 1986	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
17	16	pm5.32ri 586	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
18	17	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20		rankon 9784	. . . . . . . . . . . . . . . . 17 ⊢ (rank‘𝑥) ∈ On
21	20	oneli 6475	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22		r1fnon 9756	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
23		fndm 6638	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
25	21, 24	eleqtrrdi 2871	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26		rankr1id 9855	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝑦)) = 𝑦)
27	25, 26	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1‘𝑦)) = 𝑦)
28	27	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1‘𝑦)))
29		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝑦) ∈ V
30		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑦) → (rank‘𝑧) = (rank‘(𝑅1‘𝑦)))
31	30	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1‘𝑦))))
32	29, 31	spcev 3560	. . . . . . . . . . . . 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 558	. . . . . . . . . . 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 4016	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
40	39	unissi 4876	. . . . 5 ⊢ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪ (rank‘𝑥)
41	10, 40	eqsstri 3977	. . . 4 ⊢ (rank‘∪ 𝑥) ⊆ ∪ (rank‘𝑥)
42		pwuni 4906	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥
43		vuniex 7747	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
44	43	pwex 5345	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝑥 ∈ V
45	44	rankss 9840	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥))
46	42, 45	ax-mp 5	. . . . . . 7 ⊢ (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥)
47	43	rankpw 9833	. . . . . . 7 ⊢ (rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥)
48	46, 47	sseqtri 3979	. . . . . 6 ⊢ (rank‘𝑥) ⊆ suc (rank‘∪ 𝑥)
49	48	unissi 4876	. . . . 5 ⊢ ∪ (rank‘𝑥) ⊆ ∪ suc (rank‘∪ 𝑥)
50		rankon 9784	. . . . . 6 ⊢ (rank‘∪ 𝑥) ∈ On
51	50	onunisuci 6481	. . . . 5 ⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥)
52	49, 51	sseqtri 3979	. . . 4 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥)
53	41, 52	eqssi 3947	. . 3 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
54	5, 53	vtoclg 3517	. 2 ⊢ (𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
55		uniexb 7769	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
56		fvprc 6873	. . . . 5 ⊢ (¬ ∪ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
57	55, 56	sylnbi 333	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
58		uni0 4896	. . . 4 ⊢ ∪ ∅ = ∅
59	57, 58	eqtr4di 2813	. . 3 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ ∅)
60		fvprc 6873	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
61	60	unieqd 4880	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ (rank‘𝐴) = ∪ ∅)
62	59, 61	eqtr4d 2798	. 2 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
63	54, 62	pm2.61i 184	1 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951  dom cdm 5655  Oncon0 6359  suc csuc 6361   Fn wfn 6530  ‘cfv 6535  𝑅₁cr1 9751  rankcrnk 9752

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-reg 9571  ax-inf2 9627

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-om 7869  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-r1 9753  df-rank 9754

This theorem is used by:  rankuniss  9859  rankbnd2  9862  rankxplim2  9873  rankxplim3  9874  rankxpsuc  9875  hfuni  9890  r1limwun  10770