Theorem rankuni2b 8880

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 8846	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
2		rankval3b 8853	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
3	1, 2	sylbi 207	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
4		iuneq1 4668	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
5	4	eleq1d 2835	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On ↔ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On))
6		vex 3354	. . . . . . 7 ⊢ 𝑦 ∈ V
7		rankon 8822	. . . . . . . 8 ⊢ (rank‘𝑥) ∈ On
8	7	rgenw 3073	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
9		iunon 7589	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ ∀𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On) → ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On)
10	6, 8, 9	mp2an 672	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑦 (rank‘𝑥) ∈ On
11	5, 10	vtoclg 3417	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On)
12		eluni2 4578	. . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
13		nfv 1995	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ ∪ (𝑅1 “ On)
14		nfiu1 4684	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
15	14	nfel2 2930	. . . . . . . 8 ⊢ Ⅎ𝑥(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)
16		r1elssi 8832	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
17	16	sseld 3751	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ (𝑅1 “ On)))
18		rankelb 8851	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
19	17, 18	syl6 35	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
20		ssiun2 4697	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
21	20	sseld 3751	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
22	21	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))))
23	19, 22	syldd 72	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))))
24	13, 15, 23	rexlimd 3174	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
25	12, 24	syl5bi 232	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑦 ∈ ∪ 𝐴 → (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
26	25	ralrimiv 3114	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
27		eleq2 2839	. . . . . . 7 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
28	27	ralbidv 3135	. . . . . 6 ⊢ (𝑧 = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) → (∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
29	28	elrab 3515	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ↔ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ On ∧ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)))
30	11, 26, 29	sylanbrc 572	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧})
31		intss1 4626	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
32	30, 31	syl 17	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ ∪ 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
33	3, 32	eqsstrd 3788	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) ⊆ ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))
34	1	biimpi 206	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
35		elssuni 4603	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
36		rankssb 8875	. . . . 5 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ⊆ ∪ 𝐴 → (rank‘𝑥) ⊆ (rank‘∪ 𝐴)))
37	34, 35, 36	syl2im 40	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ⊆ (rank‘∪ 𝐴)))
38	37	ralrimiv 3114	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
39		iunss 4695	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
40	38, 39	sylibr 224	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘∪ 𝐴))
41	33, 40	eqssd 3769	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥))

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351   ⊆ wss 3723  ∪ cuni 4574  ∩ cint 4611  ∪ ciun 4654   “ cima 5252  Oncon0 5864  ‘cfv 6029  𝑅₁cr1 8789  rankcrnk 8790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792

This theorem is referenced by:  rankuni2  8882  rankcf  9801