Theorem f1finf1o 9333

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.) Avoid ax-pow 5383. (Revised by BTernaryTau, 4-Jan-2025.)

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 484	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		f1f 6817	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	adantl 481	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴⟶𝐵)
4	3	ffnd 6748	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
5		simpll 766	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵)
6	3	frnd 6755	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ⊆ 𝐵)
7		df-pss 3996	. . . . . . . . . 10 ⊢ (ran 𝐹 ⊊ 𝐵 ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≠ 𝐵))
8	7	baib 535	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ 𝐵 → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
9	6, 8	syl 17	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
10		php3 9275	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊊ 𝐵) → ran 𝐹 ≺ 𝐵)
11	10	ex 412	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
12	11	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
13		enfii 9252	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
14	13	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
15		f1f1orn 6873	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
16		f1oenfi 9245	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
17	14, 15, 16	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝐹)
18		endom 9039	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ ran 𝐹 → 𝐴 ≼ ran 𝐹)
19		domsdomtrfi 9268	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
20	18, 19	syl3an2 1164	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
21	20	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
22	14, 17, 21	syl2an2r 684	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
23	12, 22	syld 47	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → 𝐴 ≺ 𝐵))
24		sdomnen 9041	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
25	23, 24	syl6 35	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ¬ 𝐴 ≈ 𝐵))
26	9, 25	sylbird 260	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≠ 𝐵 → ¬ 𝐴 ≈ 𝐵))
27	26	necon4ad 2965	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≈ 𝐵 → ran 𝐹 = 𝐵))
28	5, 27	mpd 15	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵)
29		df-fo 6579	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
30	4, 28, 29	sylanbrc 582	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–onto→𝐵)
31		df-f1o 6580	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
32	1, 30, 31	sylanbrc 582	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
33	32	ex 412	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵))
34		f1of1 6861	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
35	33, 34	impbid1 225	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ⊆ wss 3976   ⊊ wpss 3977   class class class wbr 5166  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –1-1→wf1 6570  –onto→wfo 6571  –1-1-onto→wf1o 6572   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007

This theorem is referenced by:  hashfac  14507  crth  16825  eulerthlem2  16829  fidomndrnglem  20795  mdetunilem8  22646  basellem4  27145  lgsqrlem4  27411  lgseisenlem2  27438  aks5lem7  42157