Theorem rankr1ai 9751

Description: One direction of rankr1a 9789. (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 6895	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1val1 9739	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥))
3	2	eleq2d 2847	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥)))
4		eliun 4952	. . . . 5 ⊢ (𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
5	3, 4	bitrdi 289	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥)))
6		r1funlim 9719	. . . . . . . . . . 11 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 489	. . . . . . . . . 10 ⊢ Lim dom 𝑅1
8		limord 6401	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
10		ordtr1 6384	. . . . . . . . 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 462	. . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ dom 𝑅1)
13		r1sucg 9722	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
14	13	eleq2d 2847	. . . . . . 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 7760	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
17	9, 16	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 ⊆ On
18	17, 12	sselid 3934	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
19		rabid 3434	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20		intss1 4920	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
21	19, 20	sylbir 237	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
22	18, 21	sylan 589	. . . . . . 7 ⊢ (((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥)
23	22	ex 416	. . . . . 6 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝑅1‘suc 𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
24	15, 23	sylbird 262	. . . . 5 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
25	24	reximdva 3174	. . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝒫 (𝑅1‘𝑥) → ∃𝑥 ∈ 𝐵 ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
26	5, 25	sylbid 242	. . 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 9749	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
29		rankvalb 9750	. . . . . . 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 3967	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
32	31	adantr 484	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥))
33		rankon 9748	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
34	17, 1	sselid 3934	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
35		ontr2 6388	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵))
36	33, 34, 35	sylancr 596	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (((rank‘𝐴) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵))
37	36	expcomd 420	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝑥 ∈ 𝐵 → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵)))
38	37	imp 410	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → ((rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
39	32, 38	sylbird 262	. . 3 ⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝑥 ∈ 𝐵) → (∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ⊆ 𝑥 → (rank‘𝐴) ∈ 𝐵))
40	39	rexlimdva 3162	. 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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  ∪ cuni 4864  ∩ cint 4904  ∪ ciun 4948  dom cdm 5645   “ cima 5648  Ord word 6339  Oncon0 6340  Lim wlim 6341  suc csuc 6342  Fun wfun 6509  ‘cfv 6515  𝑅₁cr1 9715  rankcrnk 9716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-r1 9717  df-rank 9718

This theorem is referenced by:  rankr1ag  9755  tcrank  9837  dfac12lem1  10095  dfac12lem2  10096  r1limwun  10689  inatsk  10731  r1elcl  35354  r1filimi  35359  onvf1odlem4  35409  aomclem4  43587  r1rankcld  44760