Theorem f1opwfi 9320

Description: A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)

Assertion

Ref	Expression
f1opwfi	⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin))

Distinct variable groups:   𝐴,𝑏   𝐵,𝑏   𝐹,𝑏

Proof of Theorem f1opwfi

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. 2 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏))
2		simpr 490	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
3	2	elin2d 4158	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ∈ Fin)
4		f1ofun 6826	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
5		elinel1 4154	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ∈ 𝒫 𝐴)
6		elpwi 4571	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ⊆ 𝐴)
8	7	adantl 487	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ⊆ 𝐴)
9		f1odm 6828	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → dom 𝐹 = 𝐴)
11	8, 10	sseqtrrd 3975	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ⊆ dom 𝐹)
12		fores 6806	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑏 ⊆ dom 𝐹) → (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏))
13	4, 11, 12	syl2an2r 698	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏))
14		fofi 9280	. . . . 5 ⊢ ((𝑏 ∈ Fin ∧ (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏)) → (𝐹 “ 𝑏) ∈ Fin)
15	3, 13, 14	syl2anc 596	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ Fin)
16		imassrn 6075	. . . . . 6 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
17		f1ofo 6832	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
18		forn 6799	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
19	17, 18	syl 18	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 = 𝐵)
20	16, 19	sseqtrid 3980	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ⊆ 𝐵)
21	20	adantr 486	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ⊆ 𝐵)
22	15, 21	elpwd 4570	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
23	22, 15	elind 4153	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ (𝒫 𝐵 ∩ Fin))
24		simpr 490	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐵 ∩ Fin))
25	24	elin2d 4158	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
26		dff1o3 6831	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))
27	26	simprbi 503	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun ◡𝐹)
28		elinel1 4154	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝒫 𝐵 ∩ Fin) → 𝑎 ∈ 𝒫 𝐵)
29	28	adantl 487	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐵)
30		elpwi 4571	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
31	29, 30	syl 18	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ⊆ 𝐵)
32		f1ocnv 6837	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
33	32	adantr 486	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → ◡𝐹:𝐵–1-1-onto→𝐴)
34		f1odm 6828	. . . . . . . 8 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → dom ◡𝐹 = 𝐵)
35	33, 34	syl 18	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → dom ◡𝐹 = 𝐵)
36	31, 35	sseqtrrd 3975	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ⊆ dom ◡𝐹)
37		fores 6806	. . . . . 6 ⊢ ((Fun ◡𝐹 ∧ 𝑎 ⊆ dom ◡𝐹) → (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎))
38	27, 36, 37	syl2an2r 698	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎))
39		fofi 9280	. . . . 5 ⊢ ((𝑎 ∈ Fin ∧ (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎)) → (◡𝐹 “ 𝑎) ∈ Fin)
40	25, 38, 39	syl2anc 596	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ Fin)
41		imassrn 6075	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
42		dfdm4 5887	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
43	42, 9	eqtr3id 2814	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran ◡𝐹 = 𝐴)
44	41, 43	sseqtrid 3980	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
45	44	adantr 486	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
46	40, 45	elpwd 4570	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
47	46, 40	elind 4153	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin))
48	5, 28	anim12i 625	. . 3 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵))
49	30	adantl 487	. . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
50		foimacnv 6842	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
51	17, 49, 50	syl2an 608	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
52	51	eqcomd 2771	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
53		imaeq2 6060	. . . . . 6 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
54	53	eqeq2d 2776	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝑎 = (𝐹 “ 𝑏) ↔ 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎))))
55	52, 54	syl5ibrcom 250	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) → 𝑎 = (𝐹 “ 𝑏)))
56		f1of1 6823	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
57	6	adantr 486	. . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
58		f1imacnv 6841	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑏 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
59	56, 57, 58	syl2an 608	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
60	59	eqcomd 2771	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
61		imaeq2 6060	. . . . . 6 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
62	61	eqeq2d 2776	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏))))
63	60, 62	syl5ibrcom 250	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹 “ 𝑏) → 𝑏 = (◡𝐹 “ 𝑎)))
64	55, 63	impbid 215	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
65	48, 64	sylan2 605	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin))) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
66	1, 23, 47, 65	f1o2d 7674	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Fun wfun 6534  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539  Fincfn 8949

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-1o 8459  df-en 8950  df-dom 8951  df-fin 8953

This theorem is used by:  fictb  10243  ackbijnn  15907  tsmsf1o  24355  eulerpartgbij  34829