Theorem ssfiALT 8957

Description: Shorter proof of ssfi 8956 using ax-pow 5288. (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 8764	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8743	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝑥)
3		f1ofo 6723	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–onto→𝑥)
4		imassrn 5980	. . . . . . . . . . . 12 ⊢ (𝑧 “ 𝐵) ⊆ ran 𝑧
5		forn 6691	. . . . . . . . . . . 12 ⊢ (𝑧:𝐴–onto→𝑥 → ran 𝑧 = 𝑥)
6	4, 5	sseqtrid 3973	. . . . . . . . . . 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 8952	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → (𝑧 “ 𝐵) ∈ Fin)
9		isfi 8764	. . . . . . . . . . 11 ⊢ ((𝑧 “ 𝐵) ∈ Fin ↔ ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
10	8, 9	sylib 217	. . . . . . . . . 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 714	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
13		f1of1 6715	. . . . . . . . . . . . . 14 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–1-1→𝑥)
14		f1ores 6730	. . . . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
17	16	resex 5939	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ↾ 𝐵) ∈ V
18		f1oeq1 6704	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑧 ↾ 𝐵) → (𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵) ↔ (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵)))
19	17, 18	spcev 3545	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
20		bren 8743	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ≈ (𝑧 “ 𝐵) ↔ ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
21	19, 20	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → 𝐵 ≈ (𝑧 “ 𝐵))
22		entr 8792	. . . . . . . . . . . . . 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 413	. . . . . . . . . . 11 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ≈ 𝑦))
26	25	reximdv 3202	. . . . . . . . . 10 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → ∃𝑦 ∈ ω 𝐵 ≈ 𝑦))
27		isfi 8764	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ ∃𝑦 ∈ ω 𝐵 ≈ 𝑦)
28	26, 27	syl6ibr 251	. . . . . . . . 9 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
29	28	adantl 482	. . . . . . . 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 421	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
32	31	exlimdv 1936	. . . . 5 ⊢ (𝑥 ∈ ω → (∃𝑧 𝑧:𝐴–1-1-onto→𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
33	2, 32	syl5bi 241	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
34	33	rexlimiv 3209	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
35	1, 34	sylbi 216	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin))
36	35	imp 407	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074  ran crn 5590   ↾ cres 5591   “ cima 5592  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432  ωcom 7712   ≈ cen 8730  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-om 7713  df-er 8498  df-en 8734  df-fin 8737

This theorem is referenced by: (None)