Theorem f1opwfi 9053

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 2738	. 2 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)) = (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏))
2		simpr 484	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ∈ (𝒫 𝐴 ∩ Fin))
3	2	elin2d 4129	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ∈ Fin)
4		f1ofun 6702	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
5		elinel1 4125	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ∈ 𝒫 𝐴)
6		elpwi 4539	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑏 ∈ (𝒫 𝐴 ∩ Fin) → 𝑏 ⊆ 𝐴)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ⊆ 𝐴)
9		f1odm 6704	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → dom 𝐹 = 𝐴)
11	8, 10	sseqtrrd 3958	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑏 ⊆ dom 𝐹)
12		fores 6682	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑏 ⊆ dom 𝐹) → (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏))
13	4, 11, 12	syl2an2r 681	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏))
14		fofi 9035	. . . . 5 ⊢ ((𝑏 ∈ Fin ∧ (𝐹 ↾ 𝑏):𝑏–onto→(𝐹 “ 𝑏)) → (𝐹 “ 𝑏) ∈ Fin)
15	3, 13, 14	syl2anc 583	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ Fin)
16		imassrn 5969	. . . . . 6 ⊢ (𝐹 “ 𝑏) ⊆ ran 𝐹
17		f1ofo 6707	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
18		forn 6675	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
19	17, 18	syl 17	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran 𝐹 = 𝐵)
20	16, 19	sseqtrid 3969	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 “ 𝑏) ⊆ 𝐵)
21	20	adantr 480	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ⊆ 𝐵)
22	15, 21	elpwd 4538	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ 𝒫 𝐵)
23	22, 15	elind 4124	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑏 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 “ 𝑏) ∈ (𝒫 𝐵 ∩ Fin))
24		simpr 484	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ (𝒫 𝐵 ∩ Fin))
25	24	elin2d 4129	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ Fin)
26		dff1o3 6706	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))
27	26	simprbi 496	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun ◡𝐹)
28		elinel1 4125	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝒫 𝐵 ∩ Fin) → 𝑎 ∈ 𝒫 𝐵)
29	28	adantl 481	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ∈ 𝒫 𝐵)
30		elpwi 4539	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝐵 → 𝑎 ⊆ 𝐵)
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ⊆ 𝐵)
32		f1ocnv 6712	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → ◡𝐹:𝐵–1-1-onto→𝐴)
34		f1odm 6704	. . . . . . . 8 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → dom ◡𝐹 = 𝐵)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → dom ◡𝐹 = 𝐵)
36	31, 35	sseqtrrd 3958	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑎 ⊆ dom ◡𝐹)
37		fores 6682	. . . . . 6 ⊢ ((Fun ◡𝐹 ∧ 𝑎 ⊆ dom ◡𝐹) → (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎))
38	27, 36, 37	syl2an2r 681	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎))
39		fofi 9035	. . . . 5 ⊢ ((𝑎 ∈ Fin ∧ (◡𝐹 ↾ 𝑎):𝑎–onto→(◡𝐹 “ 𝑎)) → (◡𝐹 “ 𝑎) ∈ Fin)
40	25, 38, 39	syl2anc 583	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ Fin)
41		imassrn 5969	. . . . . 6 ⊢ (◡𝐹 “ 𝑎) ⊆ ran ◡𝐹
42		dfdm4 5793	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
43	42, 9	eqtr3id 2793	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran ◡𝐹 = 𝐴)
44	41, 43	sseqtrid 3969	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 “ 𝑎) ⊆ 𝐴)
45	44	adantr 480	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ⊆ 𝐴)
46	40, 45	elpwd 4538	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ 𝒫 𝐴)
47	46, 40	elind 4124	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (◡𝐹 “ 𝑎) ∈ (𝒫 𝐴 ∩ Fin))
48	5, 28	anim12i 612	. . 3 ⊢ ((𝑏 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵))
49	30	adantl 481	. . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑎 ⊆ 𝐵)
50		foimacnv 6717	. . . . . . 7 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑎 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
51	17, 49, 50	syl2an 595	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝐹 “ (◡𝐹 “ 𝑎)) = 𝑎)
52	51	eqcomd 2744	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎)))
53		imaeq2 5954	. . . . . 6 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝐹 “ 𝑏) = (𝐹 “ (◡𝐹 “ 𝑎)))
54	53	eqeq2d 2749	. . . . 5 ⊢ (𝑏 = (◡𝐹 “ 𝑎) → (𝑎 = (𝐹 “ 𝑏) ↔ 𝑎 = (𝐹 “ (◡𝐹 “ 𝑎))))
55	52, 54	syl5ibrcom 246	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) → 𝑎 = (𝐹 “ 𝑏)))
56		f1of1 6699	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
57	6	adantr 480	. . . . . . 7 ⊢ ((𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵) → 𝑏 ⊆ 𝐴)
58		f1imacnv 6716	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑏 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
59	56, 57, 58	syl2an 595	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (◡𝐹 “ (𝐹 “ 𝑏)) = 𝑏)
60	59	eqcomd 2744	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏)))
61		imaeq2 5954	. . . . . 6 ⊢ (𝑎 = (𝐹 “ 𝑏) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (𝐹 “ 𝑏)))
62	61	eqeq2d 2749	. . . . 5 ⊢ (𝑎 = (𝐹 “ 𝑏) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑏 = (◡𝐹 “ (𝐹 “ 𝑏))))
63	60, 62	syl5ibrcom 246	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑎 = (𝐹 “ 𝑏) → 𝑏 = (◡𝐹 “ 𝑎)))
64	55, 63	impbid 211	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑏 ∈ 𝒫 𝐴 ∧ 𝑎 ∈ 𝒫 𝐵)) → (𝑏 = (◡𝐹 “ 𝑎) ↔ 𝑎 = (𝐹 “ 𝑏)))
65	48, 64	sylan2 592	. 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 7501	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐹 “ 𝑏)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 𝐵 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  Fun wfun 6412  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-fin 8695

This theorem is referenced by:  fictb  9932  ackbijnn  15468  tsmsf1o  23204  eulerpartgbij  32239