Theorem rankr1ai 9683

Description: One direction of rankr1a 9721. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankr1ai	⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵)

Proof of Theorem rankr1ai

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6851	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1val1 9671	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥))
3	2	eleq2d 2815	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥)))
4		eliun 4943	. . . . 5 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
5	3, 4	bitrdi 287	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
6		r1funlim 9651	. . . . . . . . . . 11 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 485	. . . . . . . . . 10 ⊢ Lim dom 𝑅1
8		limord 6363	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
10		ordtr1 6346	. . . . . . . . 9 ⊢ (Ord dom 𝑅1 → ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
12	11	ancoms 458	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ dom 𝑅1)
13		r1sucg 9654	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
14	13	eleq2d 2815	. . . . . . 7 ⊢ (𝑥 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
15	12, 14	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
16		ordsson 7711	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
17	9, 16	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 ⊆ On
18	17, 12	sselid 3930	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
19		rabid 3414	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20		intss1 4911	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
21	19, 20	sylbir 235	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
22	18, 21	sylan 580	. . . . . . 7 ⊢ (((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
23	22	ex 412	. . . . . 6 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
24	15, 23	sylbird 260	. . . . 5 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
25	24	reximdva 3143	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
26	5, 25	sylbid 240	. . 3 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
27	1, 26	mpcom 38	. 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
28		r1elwf 9681	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
29		rankvalb 9682	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
31	30	sseq1d 3964	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
32	31	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33		rankon 9680	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
34	17, 1	sselid 3930	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
35		ontr2 6350	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵))
36	33, 34, 35	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (((rank‘𝐴) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵))
37	36	expcomd 416	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝑥 ∈ 𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
38	37	imp 406	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
39	32, 38	sylbird 260	. . 3 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → (∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
40	39	rexlimdva 3131	. 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
41	27, 40	mpd 15	1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃wrex 3054  {crab 3393   ⊆ wss 3900  𝒫 cpw 4548  ∪ cuni 4857  ∩ cint 4895  ∪ ciun 4939  dom cdm 5614   “ cima 5617  Ord word 6301  Oncon0 6302  Lim wlim 6303  suc csuc 6304  Fun wfun 6471  ‘cfv 6477  𝑅₁cr1 9647  rankcrnk 9648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9649  df-rank 9650

This theorem is referenced by:  rankr1ag  9687  tcrank  9769  dfac12lem1  10027  dfac12lem2  10028  r1limwun  10619  inatsk  10661  onvf1odlem4  35118  aomclem4  43069  r1rankcld  44243