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Theorem rankuni 9860

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 4918	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	fveq2d 6894	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘∪ 𝑥) = (rank‘∪ 𝐴))
3		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	unieqd 4921	. . . 4 ⊢ (𝑥 = 𝐴 → ∪ (rank‘𝑥) = ∪ (rank‘𝐴))
5	2, 4	eqeq12d 2746	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘∪ 𝑥) = ∪ (rank‘𝑥) ↔ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)))
6		vex 3476	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	rankuni2 9852	. . . . . 6 ⊢ (rank‘∪ 𝑥) = ∪ 𝑧 ∈ 𝑥 (rank‘𝑧)
8		fvex 6903	. . . . . . 7 ⊢ (rank‘𝑧) ∈ V
9	8	dfiun2 5035	. . . . . 6 ⊢ ∪ 𝑧 ∈ 𝑥 (rank‘𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)}
10	7, 9	eqtri 2758	. . . . 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 9836	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
13	12	anim1i 613	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
14	13	eximi 1835	. . . . . . . . 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 2819	. . . . . . . . . . . 12 ⊢ (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
17	16	pm5.32ri 574	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
18	17	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20		rankon 9792	. . . . . . . . . . . . . . . . 17 ⊢ (rank‘𝑥) ∈ On
21	20	oneli 6477	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22		r1fnon 9764	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅₁ Fn On
23		fndm 6651	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅₁ = On
25	21, 24	eleqtrrdi 2842	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅₁)
26		rankr1id 9859	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ dom 𝑅₁ ↔ (rank‘(𝑅₁‘𝑦)) = 𝑦)
27	25, 26	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅₁‘𝑦)) = 𝑦)
28	27	eqcomd 2736	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅₁‘𝑦)))
29		fvex 6903	. . . . . . . . . . . . . 14 ⊢ (𝑅₁‘𝑦) ∈ V
30		fveq2 6890	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑅₁‘𝑦) → (rank‘𝑧) = (rank‘(𝑅₁‘𝑦)))
31	30	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑅₁‘𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅₁‘𝑦))))
32	29, 31	spcev 3595	. . . . . . . . . . . . 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 547	. . . . . . . . . . 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 4066	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
40	39	unissi 4916	. . . . 5 ⊢ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑥 𝑦 = (rank‘𝑧)} ⊆ ∪ (rank‘𝑥)
41	10, 40	eqsstri 4015	. . . 4 ⊢ (rank‘∪ 𝑥) ⊆ ∪ (rank‘𝑥)
42		pwuni 4948	. . . . . . . 8 ⊢ 𝑥 ⊆ 𝒫 ∪ 𝑥
43		vuniex 7731	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
44	43	pwex 5377	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝑥 ∈ V
45	44	rankss 9846	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 ∪ 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥))
46	42, 45	ax-mp 5	. . . . . . 7 ⊢ (rank‘𝑥) ⊆ (rank‘𝒫 ∪ 𝑥)
47	43	rankpw 9840	. . . . . . 7 ⊢ (rank‘𝒫 ∪ 𝑥) = suc (rank‘∪ 𝑥)
48	46, 47	sseqtri 4017	. . . . . 6 ⊢ (rank‘𝑥) ⊆ suc (rank‘∪ 𝑥)
49	48	unissi 4916	. . . . 5 ⊢ ∪ (rank‘𝑥) ⊆ ∪ suc (rank‘∪ 𝑥)
50		rankon 9792	. . . . . 6 ⊢ (rank‘∪ 𝑥) ∈ On
51	50	onunisuci 6483	. . . . 5 ⊢ ∪ suc (rank‘∪ 𝑥) = (rank‘∪ 𝑥)
52	49, 51	sseqtri 4017	. . . 4 ⊢ ∪ (rank‘𝑥) ⊆ (rank‘∪ 𝑥)
53	41, 52	eqssi 3997	. . 3 ⊢ (rank‘∪ 𝑥) = ∪ (rank‘𝑥)
54	5, 53	vtoclg 3541	. 2 ⊢ (𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
55		uniexb 7753	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
56		fvprc 6882	. . . . 5 ⊢ (¬ ∪ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
57	55, 56	sylnbi 329	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∅)
58		uni0 4938	. . . 4 ⊢ ∪ ∅ = ∅
59	57, 58	eqtr4di 2788	. . 3 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ ∅)
60		fvprc 6882	. . . 4 ⊢ (¬ 𝐴 ∈ V → (rank‘𝐴) = ∅)
61	60	unieqd 4921	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ (rank‘𝐴) = ∪ ∅)
62	59, 61	eqtr4d 2773	. 2 ⊢ (¬ 𝐴 ∈ V → (rank‘∪ 𝐴) = ∪ (rank‘𝐴))
63	54, 62	pm2.61i 182	1 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 {cab 2707 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ∪ ciun 4996 dom cdm 5675 Oncon0 6363 suc csuc 6365 Fn wfn 6537 ‘cfv 6542 𝑅₁cr1 9759 rankcrnk 9760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762

This theorem is referenced by: rankuniss 9863 rankbnd2 9866 rankxplim2 9877 rankxplim3 9878 rankxpsuc 9879 r1limwun 10733 hfuni 35460

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