Theorem ssfiALT 9098

Description: Shorter proof of ssfi 9097 using ax-pow 5307. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ssfiALT	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Proof of Theorem ssfiALT

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 8908	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8889	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝑥)
3		f1ofo 6775	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–onto→𝑥)
4		imassrn 6026	. . . . . . . . . . . 12 ⊢ (𝑧 “ 𝐵) ⊆ ran 𝑧
5		forn 6743	. . . . . . . . . . . 12 ⊢ (𝑧:𝐴–onto→𝑥 → ran 𝑧 = 𝑥)
6	4, 5	sseqtrid 3980	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–onto→𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
7	3, 6	syl 17	. . . . . . . . . 10 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
8		ssnnfi 9093	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → (𝑧 “ 𝐵) ∈ Fin)
9		isfi 8908	. . . . . . . . . . 11 ⊢ ((𝑧 “ 𝐵) ∈ Fin ↔ ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
10	8, 9	sylib 218	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
11	7, 10	sylan2 593	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑧:𝐴–1-1-onto→𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
12	11	adantrr 717	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
13		f1of1 6767	. . . . . . . . . . . . . 14 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–1-1→𝑥)
14		f1ores 6782	. . . . . . . . . . . . . 14 ⊢ ((𝑧:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
15	13, 14	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
16		vex 3442	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
17	16	resex 5984	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ↾ 𝐵) ∈ V
18		f1oeq1 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑧 ↾ 𝐵) → (𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵) ↔ (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵)))
19	17, 18	spcev 3563	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
20		bren 8889	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ≈ (𝑧 “ 𝐵) ↔ ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
21	19, 20	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → 𝐵 ≈ (𝑧 “ 𝐵))
22		entr 8938	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≈ (𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
23	21, 22	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
24	15, 23	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
25	24	ex 412	. . . . . . . . . . 11 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ≈ 𝑦))
26	25	reximdv 3144	. . . . . . . . . 10 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → ∃𝑦 ∈ ω 𝐵 ≈ 𝑦))
27		isfi 8908	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ ∃𝑦 ∈ ω 𝐵 ≈ 𝑦)
28	26, 27	imbitrrdi 252	. . . . . . . . 9 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
29	28	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
30	12, 29	mpd 15	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ Fin)
31	30	exp32 420	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
32	31	exlimdv 1933	. . . . 5 ⊢ (𝑥 ∈ ω → (∃𝑧 𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
33	2, 32	biimtrid 242	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
34	33	rexlimiv 3123	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
35	1, 34	sylbi 217	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
36	35	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  ran crn 5624   ↾ cres 5625   “ cima 5626  –1-1→wf1 6483  –onto→wfo 6484  –1-1-onto→wf1o 6485  ωcom 7806   ≈ cen 8876  Fincfn 8879

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 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-om 7807  df-er 8632  df-en 8880  df-fin 8883

This theorem is referenced by: (None)