Theorem rankval3b 9750

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, 17-Nov-2014.)

Assertion

Ref	Expression
rankval3b	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem rankval3b

Step	Hyp	Ref	Expression
1		rankon 9719	. . . . . . . . . 10 ⊢ (rank‘𝐴) ∈ On
2		simprl 771	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3		ontri1 6358	. . . . . . . . . 10 ⊢ (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
4	1, 2, 3	sylancr 588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
5	4	con2bid 354	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6		r1elssi 9729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
7	6	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ ∪ (𝑅1 “ On))
8	7	sselda 3922	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ (𝑅1 “ On))
9		rankdmr1 9725	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘𝐴) ∈ dom 𝑅1
10		r1funlim 9690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
11	10	simpri 485	. . . . . . . . . . . . . . . . . . 19 ⊢ Lim dom 𝑅1
12		limord 6385	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
13		ordtr1 6368	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
14	11, 12, 13	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
15	9, 14	mpan2 692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
16	15	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ dom 𝑅1)
17		rankr1ag 9726	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
18	8, 16, 17	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ (𝑅1‘𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
19	18	ralbidva 3159	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥))
20	19	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
21	20	an32s 653	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
22		dfss3 3911	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ (𝑅1‘𝑥))
23	21, 22	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘𝑥))
24		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ∈ ∪ (𝑅1 “ On))
25	15	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26		rankr1bg 9727	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
27	24, 25, 26	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1‘𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
28	23, 27	mpbid 232	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
29	28	ex 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
30	29	adantrl 717	. . . . . . . 8 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
31	5, 30	sylbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
32	31	pm2.18d 127	. . . . . 6 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
33	32	ex 412	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
34	33	alrimiv 1929	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35		ssintab 4908	. . . 4 ⊢ ((rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
36	34, 35	sylibr 234	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)})
37		df-rab 3391	. . . 4 ⊢ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
38	37	inteqi 4894	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} = ∩ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥)}
39	36, 38	sseqtrrdi 3964	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) ⊆ ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})
40		rankelb 9748	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
41	40	ralrimiv 3129	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42		eleq2 2826	. . . . 5 ⊢ (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
43	42	ralbidv 3161	. . . 4 ⊢ (𝑥 = (rank‘𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
44	43	onintss 6376	. . 3 ⊢ ((rank‘𝐴) ∈ On → (∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ (rank‘𝐴) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴)))
45	1, 41, 44	mpsyl 68	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴))
46	39, 45	eqssd 3940	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3390   ⊆ wss 3890  ∪ cuni 4851  ∩ cint 4890  dom cdm 5631   “ cima 5634  Ord word 6323  Oncon0 6324  Lim wlim 6325  Fun wfun 6493  ‘cfv 6499  𝑅₁cr1 9686  rankcrnk 9687

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-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

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 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-rank 9689

This theorem is referenced by:  ranksnb  9751  rankonidlem  9752  rankval3  9764  rankunb  9774  rankuni2b  9777  tcrank  9808