Theorem rankuni2b 9824

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 9790	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		rankval3b 9797	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
3	1, 2	sylbi 220	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
4		eleq2 2850	. . . . . 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 4972	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
7	6	eleq1d 2846	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On))
8		vex 3457	. . . . . . 7 ⊢ 𝑦 ∈ V
9		rankon 9766	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
10	9	rgenw 3081	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
11		iunon 8325	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On)
12	8, 10, 11	mp2an 704	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
13	7, 12	vtoclg 3521	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On)
14		eluni2 4875	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
15		nfv 1942	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ ∪ (𝑅1 “ On)
16		nfiu1 4991	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
17	16	nfel2 2941	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
18		r1elssi 9776	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
19	18	sseld 3935	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ On)))
20		rankelb 9795	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
21	19, 20	syl6 36	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
22		ssiun2 5011	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
23	22	sseld 3935	. . . . . . . . . 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 73	. . . . . . . 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 245	. . . . . 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 3651	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
30		intss1 4927	. . . 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 18	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
32	3, 31	eqsstrd 3970	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
33	1	biimpi 219	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
34		elssuni 4903	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
35		rankssb 9819	. . . . 5 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ⊆ ∪ 𝐴 → (rank‘𝑥) ⊆ (rank‘∪ 𝐴)))
36	33, 34, 35	syl2im 41	. . . 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 5008	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
39	37, 38	sylibr 237	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
40	32, 39	eqssd 3953	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  {crab 3414  Vcvv 3453   ⊆ wss 3904  ∪ cuni 4871  ∩ cint 4911  ∪ ciun 4955   “ cima 5664  Oncon0 6360  ‘cfv 6536  𝑅₁cr1 9733  rankcrnk 9734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9735  df-rank 9736

This theorem is referenced by:  rankuni2  9826  rankcf  10761