Theorem rankuni2b 9611

Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.)

Assertion

Ref	Expression
rankuni2b	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankuni2b

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

Step	Hyp	Ref	Expression
1		uniwf 9577	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		rankval3b 9584	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
3	1, 2	sylbi 216	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
4		eleq2 2827	. . . . . 6 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
5	4	ralbidv 3112	. . . . 5 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
6		iuneq1 4940	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
7	6	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On))
8		vex 3436	. . . . . . 7 ⊢ 𝑦 ∈ V
9		rankon 9553	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
10	9	rgenw 3076	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
11		iunon 8170	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On)
12	8, 10, 11	mp2an 689	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
13	7, 12	vtoclg 3505	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On)
14		eluni2 4843	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
15		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ ∪ (𝑅1 “ On)
16		nfiu1 4958	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
17	16	nfel2 2925	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
18		r1elssi 9563	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
19	18	sseld 3920	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ On)))
20		rankelb 9582	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
21	19, 20	syl6 35	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
22		ssiun2 4977	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
23	22	sseld 3920	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))))
25	21, 24	syldd 72	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))))
26	15, 17, 25	rexlimd 3250	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
27	14, 26	syl5bi 241	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ ∪ 𝐴 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
28	27	ralrimiv 3102	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
29	5, 13, 28	elrabd 3626	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
30		intss1 4894	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
31	29, 30	syl 17	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
32	3, 31	eqsstrd 3959	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
33	1	biimpi 215	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
34		elssuni 4871	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
35		rankssb 9606	. . . . 5 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ⊆ ∪ 𝐴 → (rank‘𝑥) ⊆ (rank‘∪ 𝐴)))
36	33, 34, 35	syl2im 40	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘∪ 𝐴)))
37	36	ralrimiv 3102	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
38		iunss 4975	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
39	37, 38	sylibr 233	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
40	32, 39	eqssd 3938	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ∩ cint 4879  ∪ ciun 4924   “ cima 5592  Oncon0 6266  ‘cfv 6433  𝑅₁cr1 9520  rankcrnk 9521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-r1 9522  df-rank 9523

This theorem is referenced by:  rankuni2  9613  rankcf  10533