Theorem ssfiALT 9193

Description: Shorter proof of ssfi 9192 using ax-pow 5326. (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 9006	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8987	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝑥)
3		f1ofo 6833	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–onto→𝑥)
4		imassrn 6197	. . . . . . . . . . . 12 ⊢ (𝑧 “ 𝐵) ⊆ ran 𝑧
5		forn 6800	. . . . . . . . . . . 12 ⊢ (𝑧:𝐴–onto→𝑥 → ran 𝑧 = 𝑥)
6	4, 5	sseqtrid 3950	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–onto→𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
7	3, 6	syl 18	. . . . . . . . . 10 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
8		ssnnfi 9189	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → (𝑧 “ 𝐵) ∈ Fin)
9		isfi 9006	. . . . . . . . . . 11 ⊢ ((𝑧 “ 𝐵) ∈ Fin ↔ ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
10	8, 9	sylib 221	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
11	7, 10	sylan2 605	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑧:𝐴–1-1-onto→𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
12	11	adantrr 730	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
13		f1of1 6824	. . . . . . . . . . . . . 14 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–1-1→𝑥)
14		f1ores 6840	. . . . . . . . . . . . . 14 ⊢ ((𝑧:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
15	13, 14	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
16		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
17	16	resex 6017	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ↾ 𝐵) ∈ V
18		f1oeq1 6813	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑧 ↾ 𝐵) → (𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵) ↔ (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵)))
19	17, 18	spcev 3560	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
20		bren 8987	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ≈ (𝑧 “ 𝐵) ↔ ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
21	19, 20	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → 𝐵 ≈ (𝑧 “ 𝐵))
22		entr 9037	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≈ (𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
23	21, 22	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
24	15, 23	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
25	24	ex 418	. . . . . . . . . . 11 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ≈ 𝑦))
26	25	reximdv 3177	. . . . . . . . . 10 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → ∃𝑦 ∈ ω 𝐵 ≈ 𝑦))
27		isfi 9006	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ ∃𝑦 ∈ ω 𝐵 ≈ 𝑦)
28	26, 27	imbitrrdi 255	. . . . . . . . 9 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
29	28	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
30	12, 29	mpd 16	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ Fin)
31	30	exp32 426	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
32	31	exlimdv 1965	. . . . 5 ⊢ (𝑥 ∈ ω → (∃𝑧 𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
33	2, 32	biimtrid 245	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
34	33	rexlimiv 3156	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
35	1, 34	sylbi 220	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
36	35	imp 412	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   class class class wbr 5102  ran crn 5651   ↾ cres 5652   “ cima 5653  –1-1→wf1 6536  –onto→wfo 6537  –1-1-onto→wf1o 6538  ωcom 7878   ≈ cen 8974  Fincfn 8977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-om 7879  df-er 8721  df-en 8978  df-fin 8981

This theorem is used by: (None)