Theorem rankuni2b 9809

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 9775	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		rankval3b 9782	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
3	1, 2	sylbi 219	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
4		eleq2 2852	. . . . . 6 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
5	4	ralbidv 3186	. . . . 5 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
6		iuneq1 4967	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
7	6	eleq1d 2848	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On))
8		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
9		rankon 9751	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
10	9	rgenw 3081	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
11		iunon 8310	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On)
12	8, 10, 11	mp2an 702	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
13	7, 12	vtoclg 3523	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On)
14		eluni2 4870	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
15		nfv 1935	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ ∪ (𝑅1 “ On)
16		nfiu1 4986	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
17	16	nfel2 2943	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
18		r1elssi 9761	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
19	18	sseld 3936	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ On)))
20		rankelb 9780	. . . . . . . . . 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 5006	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
23	22	sseld 3936	. . . . . . . . . 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 3270	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
27	14, 26	biimtrid 244	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ ∪ 𝐴 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
28	27	ralrimiv 3154	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
29	5, 13, 28	elrabd 3653	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
30		intss1 4922	. . . 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 3971	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
33	1	biimpi 218	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
34		elssuni 4898	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
35		rankssb 9804	. . . . 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 3154	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
38		iunss 5003	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
39	37, 38	sylibr 236	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
40	32, 39	eqssd 3954	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087  {crab 3415  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4866  ∩ cint 4906  ∪ ciun 4950   “ cima 5651  Oncon0 6346  ‘cfv 6521  𝑅₁cr1 9718  rankcrnk 9719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-r1 9720  df-rank 9721

This theorem is referenced by:  rankuni2  9811  rankcf  10746