Theorem f1finf1o 7137

Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)

Assertion

Ref	Expression
f1finf1o	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

Proof of Theorem f1finf1o

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		simplr 528	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐵 ∈ Fin)
3		f1rn 5540	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
4	3	adantl 277	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ⊆ 𝐵)
5		f1fn 5541	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
6		fnima 5448	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
7	5, 6	syl 14	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐴) = ran 𝐹)
8	7	adantl 277	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 “ 𝐴) = ran 𝐹)
9		ssid 3245	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
10	9	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ⊆ 𝐴)
11		simpll 527	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵)
12		enfii 7056	. . . . . . . . 9 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
13	2, 11, 12	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin)
14		f1imaeng 6961	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 “ 𝐴) ≈ 𝐴)
15	1, 10, 13, 14	syl3anc 1271	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 “ 𝐴) ≈ 𝐴)
16	8, 15	eqbrtrrd 4110	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ≈ 𝐴)
17		entr 6953	. . . . . 6 ⊢ ((ran 𝐹 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → ran 𝐹 ≈ 𝐵)
18	16, 11, 17	syl2anc 411	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ≈ 𝐵)
19		fisseneq 7119	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≈ 𝐵) → ran 𝐹 = 𝐵)
20	2, 4, 18, 19	syl3anc 1271	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵)
21		dff1o5 5589	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))
22	1, 20, 21	sylanbrc 417	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
23	22	ex 115	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵))
24		f1of1 5579	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
25	23, 24	impbid1 142	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ⊆ wss 3198   class class class wbr 4086  ran crn 4724   “ cima 4726   Fn wfn 5319  –1-1→wf1 5321  –1-1-onto→wf1o 5323   ≈ cen 6902  Fincfn 6904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by:  iseqf1olemqf1o  10758  crth  12786  eulerthlemh  12793  lgseisenlem2  15790  pwf1oexmid  16536