Theorem f1finf1o 8342

Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)

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 471	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		f1f 6241	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	adantl 467	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴⟶𝐵)
4		ffn 6185	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5	3, 4	syl 17	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
6		simpll 742	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵)
7		frn 6193	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
8	3, 7	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ⊆ 𝐵)
9		df-pss 3739	. . . . . . . . . 10 ⊢ (ran 𝐹 ⊊ 𝐵 ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≠ 𝐵))
10	9	baib 517	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ 𝐵 → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
11	8, 10	syl 17	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
12		simplr 744	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐵 ∈ Fin)
13		relen 8113	. . . . . . . . . . . . . . 15 ⊢ Rel ≈
14	13	brrelexi 5298	. . . . . . . . . . . . . 14 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
15	6, 14	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ V)
16	12, 15	elmapd 8022	. . . . . . . . . . . 12 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝐹:𝐴⟶𝐵))
17	3, 16	mpbird 247	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ (𝐵 ↑𝑚 𝐴))
18		f1f1orn 6289	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
19	18	adantl 467	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→ran 𝐹)
20		f1oen3g 8124	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
21	17, 19, 20	syl2anc 565	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝐹)
22		php3 8301	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊊ 𝐵) → ran 𝐹 ≺ 𝐵)
23	22	ex 397	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
24	12, 23	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
25		ensdomtr 8251	. . . . . . . . . 10 ⊢ ((𝐴 ≈ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
26	21, 24, 25	syl6an 655	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → 𝐴 ≺ 𝐵))
27		sdomnen 8137	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
28	26, 27	syl6 35	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ¬ 𝐴 ≈ 𝐵))
29	11, 28	sylbird 250	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≠ 𝐵 → ¬ 𝐴 ≈ 𝐵))
30	29	necon4ad 2962	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≈ 𝐵 → ran 𝐹 = 𝐵))
31	6, 30	mpd 15	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵)
32		df-fo 6037	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
33	5, 31, 32	sylanbrc 564	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–onto→𝐵)
34		df-f1o 6038	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
35	1, 33, 34	sylanbrc 564	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
36	35	ex 397	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵))
37		f1of1 6277	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
38	36, 37	impbid1 215	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  Vcvv 3351   ⊆ wss 3723   ⊊ wpss 3724   class class class wbr 4786  ran crn 5250   Fn wfn 6026  ⟶wf 6027  –1-1→wf1 6028  –onto→wfo 6029  –1-1-onto→wf1o 6030  (class class class)co 6792   ↑_𝑚 cmap 8008   ≈ cen 8105   ≺ csdm 8107  Fincfn 8108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112

This theorem is referenced by:  hashfac  13443  crth  15689  eulerthlem2  15693  fidomndrnglem  19520  mdetunilem8  20642  basellem4  25030  lgsqrlem4  25294  lgseisenlem2  25321