Theorem ssfiALT 9227

Description: Shorter proof of ssfi 9226 using ax-pow 5334. (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 9040	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 9021	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑧 𝑧:𝐴⟶1-1onto𝑥)
3		f1ofo 6862	. . . . . . . . . . 11 ⊢ (𝑧:𝐴⟶1-1onto𝑥 → 𝑧:𝐴⟶onto𝑥)
4		imassrn 6216	. . . . . . . . . . . 12 ⊢ (𝑧 “ 𝐵) ⊆ ran 𝑧
5		forn 6829	. . . . . . . . . . . 12 ⊢ (𝑧:𝐴⟶onto𝑥 → ran 𝑧 = 𝑥)
6	4, 5	sseqtrid 3955	. . . . . . . . . . 11 ⊢ (𝑧:𝐴⟶onto𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
7	3, 6	syl 18	. . . . . . . . . 10 ⊢ (𝑧:𝐴⟶1-1onto𝑥 → (𝑧 “ 𝐵) ⊆ 𝑥)
8		ssnnfi 9223	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → (𝑧 “ 𝐵) ∈ Fin)
9		isfi 9040	. . . . . . . . . . 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⟶1-1onto𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
12	11	adantrr 730	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴)) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
13		f1of1 6853	. . . . . . . . . . . . . 14 ⊢ (𝑧:𝐴⟶1-1onto𝑥 → 𝑧:𝐴⟶1-1𝑥)
14		f1ores 6869	. . . . . . . . . . . . . 14 ⊢ ((𝑧:𝐴⟶1-1𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵))
15	13, 14	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵))
16		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
17	16	resex 6036	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ↾ 𝐵) ∈ V
18		f1oeq1 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑧 ↾ 𝐵) → (𝑥:𝐵⟶1-1onto(𝑧 “ 𝐵) ↔ (𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵)))
19	17, 18	spcev 3564	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵) → ∃𝑥 𝑥:𝐵⟶1-1onto(𝑧 “ 𝐵))
20		bren 9021	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ≈ (𝑧 “ 𝐵) ↔ ∃𝑥 𝑥:𝐵⟶1-1onto(𝑧 “ 𝐵))
21	19, 20	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵) → 𝐵 ≈ (𝑧 “ 𝐵))
22		entr 9071	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≈ (𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
23	21, 22	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝑧 ↾ 𝐵):𝐵⟶1-1onto(𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
24	15, 23	sylan 592	. . . . . . . . . . . 12 ⊢ (((𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
25	24	ex 418	. . . . . . . . . . 11 ⊢ ((𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ≈ 𝑦))
26	25	reximdv 3181	. . . . . . . . . 10 ⊢ ((𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → ∃𝑦 ∈ ω 𝐵 ≈ 𝑦))
27		isfi 9040	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin ↔ ∃𝑦 ∈ ω 𝐵 ≈ 𝑦)
28	26, 27	imbitrrdi 255	. . . . . . . . 9 ⊢ ((𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
29	28	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴)) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → 𝐵 ∈ Fin))
30	12, 29	mpd 16	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴⟶1-1onto𝑥 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ Fin)
31	30	exp32 426	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝑧:𝐴⟶1-1onto𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
32	31	exlimdv 1968	. . . . 5 ⊢ (𝑥 ∈ ω → (∃𝑧 𝑧:𝐴⟶1-1onto𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
33	2, 32	biimtrid 245	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐵 ⊆ 𝐴 → 𝐵 ∈ Fin)))
34	33	rexlimiv 3160	. . 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 1812   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108  ran crn 5659   ↾ cres 5660   “ cima 5661  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ωcom 7911   ≈ cen 9008  Fincfn 9011

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-om 7912  df-er 8755  df-en 9012  df-fin 9015

This theorem is used by: (None)