Theorem rankr1ai 9758

Description: One direction of rankr1a 9796. (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 6898	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1val1 9746	. . . . . 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 4962	. . . . 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 9726	. . . . . . . . . . 11 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 485	. . . . . . . . . 10 ⊢ Lim dom 𝑅1
8		limord 6396	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
10		ordtr1 6379	. . . . . . . . 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 9729	. . . . . . . 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 7762	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
17	9, 16	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 ⊆ On
18	17, 12	sselid 3947	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
19		rabid 3430	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20		intss1 4930	. . . . . . . . 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 3147	. . . 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 9756	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
29		rankvalb 9757	. . . . . . 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 3981	. . . . 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 9755	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
34	17, 1	sselid 3947	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
35		ontr2 6383	. . . . . . 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 3135	. 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 1540   ∈ wcel 2109  ∃wrex 3054  {crab 3408   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913  ∪ ciun 4958  dom cdm 5641   “ cima 5644  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  Fun wfun 6508  ‘cfv 6514  𝑅₁cr1 9722  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725

This theorem is referenced by:  rankr1ag  9762  tcrank  9844  dfac12lem1  10104  dfac12lem2  10105  r1limwun  10696  inatsk  10738  onvf1odlem4  35100  aomclem4  43053  r1rankcld  44227