Theorem rankuni2b 9767

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 9733	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		rankval3b 9740	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
3	1, 2	sylbi 217	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
4		eleq2 2824	. . . . . 6 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
5	4	ralbidv 3158	. . . . 5 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
6		iuneq1 4962	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
7	6	eleq1d 2820	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On))
8		vex 3443	. . . . . . 7 ⊢ 𝑦 ∈ V
9		rankon 9709	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
10	9	rgenw 3054	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
11		iunon 8271	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On)
12	8, 10, 11	mp2an 693	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
13	7, 12	vtoclg 3510	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On)
14		eluni2 4866	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
15		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ ∪ (𝑅1 “ On)
16		nfiu1 4981	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
17	16	nfel2 2916	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
18		r1elssi 9719	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
19	18	sseld 3931	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ On)))
20		rankelb 9738	. . . . . . . . . 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 5002	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
23	22	sseld 3931	. . . . . . . . . 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 3242	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
27	14, 26	biimtrid 242	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ ∪ 𝐴 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
28	27	ralrimiv 3126	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
29	5, 13, 28	elrabd 3647	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
30		intss1 4917	. . . 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 3967	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
33	1	biimpi 216	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
34		elssuni 4893	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
35		rankssb 9762	. . . . 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 3126	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
38		iunss 4999	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
39	37, 38	sylibr 234	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
40	32, 39	eqssd 3950	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059  {crab 3398  Vcvv 3439   ⊆ wss 3900  ∪ cuni 4862  ∩ cint 4901  ∪ ciun 4945   “ cima 5626  Oncon0 6316  ‘cfv 6491  𝑅₁cr1 9676  rankcrnk 9677

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9678  df-rank 9679

This theorem is referenced by:  rankuni2  9769  rankcf  10690