Theorem rankr1ai 9714

Description: One direction of rankr1a 9752. (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 6869	. . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1)
2		r1val1 9702	. . . . . 6 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥))
3	2	eleq2d 2823	. . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝒫 (𝑅1‘𝑥)))
4		eliun 4951	. . . . 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 9682	. . . . . . . . . . 11 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
7	6	simpri 485	. . . . . . . . . 10 ⊢ Lim dom 𝑅1
8		limord 6379	. . . . . . . . . 10 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ Ord dom 𝑅1
10		ordtr1 6362	. . . . . . . . 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 9685	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
14	13	eleq2d 2823	. . . . . . 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 7730	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
17	9, 16	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 ⊆ On
18	17, 12	sselid 3932	. . . . . . . 8 ⊢ ((𝐵 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
19		rabid 3421	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} ↔ (𝑥 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝑥)))
20		intss1 4919	. . . . . . . . 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 581	. . . . . . 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 3150	. . . 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 9712	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
29		rankvalb 9713	. . . . . . 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 3966	. . . . 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 9711	. . . . . . 7 ⊢ (rank‘𝐴) ∈ On
34	17, 1	sselid 3932	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On)
35		ontr2 6366	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ 𝐵 ∈ On) → (((rank‘𝐴) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (rank‘𝐴) ∈ 𝐵))
36	33, 34, 35	sylancr 588	. . . . . 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 3138	. 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 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3400   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864  ∩ cint 4903  ∪ ciun 4947  dom cdm 5625   “ cima 5628  Ord word 6317  Oncon0 6318  Lim wlim 6319  suc csuc 6320  Fun wfun 6487  ‘cfv 6493  𝑅₁cr1 9678  rankcrnk 9679

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680  df-rank 9681

This theorem is referenced by:  rankr1ag  9718  tcrank  9800  dfac12lem1  10058  dfac12lem2  10059  r1limwun  10651  inatsk  10693  r1elcl  35235  r1filimi  35240  onvf1odlem4  35281  aomclem4  43335  r1rankcld  44508