Theorem rankr1ai 9795

Description: One direction of rankr1a 9833. (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 6928	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1val1 9783	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥))
3	2	eleq2d 2819	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥)))
4		eliun 5001	. . . . 5 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
5	3, 4	bitrdi 286	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
6		r1funlim 9763	. . . . . . . . . . 11 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 486	. . . . . . . . . 10 ⊢ Lim dom 𝑅1
8		limord 6424	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
10		ordtr1 6407	. . . . . . . . 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 459	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ dom 𝑅1)
13		r1sucg 9766	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
14	13	eleq2d 2819	. . . . . . 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 7772	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
17	9, 16	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 ⊆ On
18	17, 12	sselid 3980	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
19		rabid 3452	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20		intss1 4967	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
21	19, 20	sylbir 234	. . . . . . . 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 413	. . . . . 6 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
24	15, 23	sylbird 259	. . . . 5 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
25	24	reximdva 3168	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
26	5, 25	sylbid 239	. . 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 9793	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
29		rankvalb 9794	. . . . . . 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 4013	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
32	31	adantr 481	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33		rankon 9792	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
34	17, 1	sselid 3980	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
35		ontr2 6411	. . . . . . 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 417	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝑥 ∈ 𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
38	37	imp 407	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
39	32, 38	sylbird 259	. . 3 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → (∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
40	39	rexlimdva 3155	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432   ⊆ wss 3948  𝒫 cpw 4602  ∪ cuni 4908  ∩ cint 4950  ∪ ciun 4997  dom cdm 5676   “ cima 5679  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  Fun wfun 6537  ‘cfv 6543  𝑅₁cr1 9759  rankcrnk 9760

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-r1 9761  df-rank 9762

This theorem is referenced by:  rankr1ag  9799  tcrank  9881  dfac12lem1  10140  dfac12lem2  10141  r1limwun  10733  inatsk  10775  aomclem4  41881  r1rankcld  43072