Theorem rankuni 9779

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 4875	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	fveq2d 6839	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) = (rank‘∪ 𝐴))
3		fveq2 6835	. . . . 5 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	unieqd 4877	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ (rank‘𝑥) = ∪ (rank‘𝐴))
5	2, 4	eqeq12d 2753	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) = ∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)))
6		vex 3445	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankuni2 9771	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧)
8		fvex 6848	. . . . . . 7 ⊢ (rank‘𝑧) ∈ V
9	8	dfiun2 4988	. . . . . 6 ⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
10	7, 9	eqtri 2760	. . . . 5 ⊢ (rank‘∪ 𝑥) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
11		df-rex 3062	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)))
12	6	rankel 9755	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
13	12	anim1i 616	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
14	13	eximi 1837	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15		19.42v 1955	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16		eleq1 2825	. . . . . . . . . . . 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 1850	. . . . . . . . . 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 9711	. . . . . . . . . . . . . . . . 17 ⊢ (rank‘𝑥) ∈ On
21	20	oneli 6433	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22		r1fnon 9683	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
23		fndm 6596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
25	21, 24	eleqtrrdi 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26		rankr1id 9778	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝑦)) = 𝑦)
27	25, 26	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1‘𝑦)) = 𝑦)
28	27	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1‘𝑦)))
29		fvex 6848	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘𝑦) ∈ V
30		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅1‘𝑦) → (rank‘𝑧) = (rank‘(𝑅1‘𝑦)))
31	30	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅1‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1‘𝑦))))
32	29, 31	spcev 3561	. . . . . . . . . . . . 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 209	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
36	15, 18, 35	3bitr3i 301	. . . . . . . . 9 ⊢ (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
37	14, 36	sylib 218	. . . . . . . 8 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
38	11, 37	sylbi 217	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
39	38	abssi 4021	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
40	39	unissi 4873	. . . . 5 ⊢ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪ (rank‘𝑥)
41	10, 40	eqsstri 3981	. . . 4 ⊢ (rank‘∪ 𝑥) ⊆ ∪ (rank‘𝑥)
42		pwuni 4902	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥
43		vuniex 7686	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
44	43	pwex 5326	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝑥 ∈ V
45	44	rankss 9765	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥))
46	42, 45	ax-mp 5	. . . . . . 7 ⊢ (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥)
47	43	rankpw 9759	. . . . . . 7 ⊢ (rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥)
48	46, 47	sseqtri 3983	. . . . . 6 ⊢ (rank‘𝑥) ⊆ suc (rank‘∪ 𝑥)
49	48	unissi 4873	. . . . 5 ⊢ ∪ (rank‘𝑥) ⊆ ∪ suc (rank‘∪ 𝑥)
50		rankon 9711	. . . . . 6 ⊢ (rank‘∪ 𝑥) ∈ On
51	50	onunisuci 6439	. . . . 5 ⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥)
52	49, 51	sseqtri 3983	. . . 4 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥)
53	41, 52	eqssi 3951	. . 3 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
54	5, 53	vtoclg 3512	. 2 ⊢ (𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
55		uniexb 7711	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
56		fvprc 6827	. . . . 5 ⊢ (¬ ∪ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
57	55, 56	sylnbi 330	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
58		uni0 4892	. . . 4 ⊢ ∪ ∅ = ∅
59	57, 58	eqtr4di 2790	. . 3 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ ∅)
60		fvprc 6827	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
61	60	unieqd 4877	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ (rank‘𝐴) = ∪ ∅)
62	59, 61	eqtr4d 2775	. 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∃wrex 3061  Vcvv 3441   ⊆ wss 3902  ∅c0 4286  𝒫 cpw 4555  ∪ cuni 4864  ∪ ciun 4947  dom cdm 5625  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  𝑅₁cr1 9678  rankcrnk 9679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680  df-rank 9681

This theorem is referenced by:  rankuniss  9782  rankbnd2  9785  rankxplim2  9796  rankxplim3  9797  rankxpsuc  9798  r1limwun  10651  hfuni  36380