Theorem f1finf1o 6963

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 5436	. . . . . 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 5437	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
6		fnima 5348	. . . . . . . . 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 3189	. . . . . . . . 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 6891	. . . . . . . . 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 6809	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 “ 𝐴) ≈ 𝐴)
15	1, 10, 13, 14	syl3anc 1248	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 “ 𝐴) ≈ 𝐴)
16	8, 15	eqbrtrrd 4041	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ≈ 𝐴)
17		entr 6801	. . . . . 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 6948	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≈ 𝐵) → ran 𝐹 = 𝐵)
20	2, 4, 18, 19	syl3anc 1248	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵)
21		dff1o5 5484	. . . 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 5474	. 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 1363   ∈ wcel 2159   ⊆ wss 3143   class class class wbr 4017  ran crn 4641   “ cima 4643   Fn wfn 5225  –1-1→wf1 5227  –1-1-onto→wf1o 5229   ≈ cen 6755  Fincfn 6757

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-1o 6434  df-er 6552  df-en 6758  df-fin 6760

This theorem is referenced by:  iseqf1olemqf1o  10510  crth  12241  eulerthlemh  12248  lgseisenlem2  14834  pwf1oexmid  15133