Theorem ssfiALT 9110

Description: Shorter proof of ssfi 9109 using ax-pow 5312. (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 8924	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		bren 8905	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝑥)
3		f1ofo 6789	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–onto→𝑥)
4		imassrn 6038	. . . . . . . . . . . 12 ⊢ (𝑧 “ 𝐵) ⊆ ran 𝑧
5		forn 6757	. . . . . . . . . . . 12 ⊢ (𝑧:𝐴–onto→𝑥 → ran 𝑧 = 𝑥)
6	4, 5	sseqtrid 3978	. . . . . . . . . . 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 9106	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ (𝑧 “ 𝐵) ⊆ 𝑥) → (𝑧 “ 𝐵) ∈ Fin)
9		isfi 8924	. . . . . . . . . . 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 594	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑧:𝐴–1-1-onto→𝑥) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
12	11	adantrr 718	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ (𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴)) → ∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦)
13		f1of1 6781	. . . . . . . . . . . . . 14 ⊢ (𝑧:𝐴–1-1-onto→𝑥 → 𝑧:𝐴–1-1→𝑥)
14		f1ores 6796	. . . . . . . . . . . . . 14 ⊢ ((𝑧:𝐴–1-1→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
15	13, 14	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵))
16		vex 3446	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
17	16	resex 5996	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ↾ 𝐵) ∈ V
18		f1oeq1 6770	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑧 ↾ 𝐵) → (𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵) ↔ (𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵)))
19	17, 18	spcev 3562	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) → ∃𝑥 𝑥:𝐵–1-1-onto→(𝑧 “ 𝐵))
20		bren 8905	. . . . . . . . . . . . . . 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 8955	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≈ (𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
23	21, 22	sylan 581	. . . . . . . . . . . . 13 ⊢ (((𝑧 ↾ 𝐵):𝐵–1-1-onto→(𝑧 “ 𝐵) ∧ (𝑧 “ 𝐵) ≈ 𝑦) → 𝐵 ≈ 𝑦)
24	15, 23	sylan 581	. . . . . . . . . . . 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 3153	. . . . . . . . . 10 ⊢ ((𝑧:𝐴–1-1-onto→𝑥 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ ω (𝑧 “ 𝐵) ≈ 𝑦 → ∃𝑦 ∈ ω 𝐵 ≈ 𝑦))
27		isfi 8924	. . . . . . . . . 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 1935	. . . . 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 3132	. . 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 1781   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100  ran crn 5633   ↾ cres 5634   “ cima 5635  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499  ωcom 7818   ≈ cen 8892  Fincfn 8895

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-om 7819  df-er 8645  df-en 8896  df-fin 8899

This theorem is referenced by: (None)