Theorem f1finf1o 9247

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 5334. (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 490	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		f1f 6775	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	adantl 487	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴⟶𝐵)
4	3	ffnd 6707	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
5		simpll 779	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵)
6	3	frnd 6715	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ⊆ 𝐵)
7		df-pss 3922	. . . . . . . . . 10 ⊢ (ran 𝐹 ⊊ 𝐵 ↔ (ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≠ 𝐵))
8	7	baib 545	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ 𝐵 → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
9	6, 8	syl 18	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
10		php3 9207	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊊ 𝐵) → ran 𝐹 ≺ 𝐵)
11	10	ex 418	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
12	11	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
13		enfii 9184	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
14	13	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
15		f1f1orn 6833	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
16		f1oenfi 9177	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
17	14, 15, 16	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝐹)
18		endom 8989	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ ran 𝐹 → 𝐴 ≼ ran 𝐹)
19		domsdomtrfi 9200	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
20	18, 19	syl3an2 1182	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
21	20	3expia 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
22	14, 17, 21	syl2an2r 698	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
23	12, 22	syld 48	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → 𝐴 ≺ 𝐵))
24		sdomnen 8991	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
25	23, 24	syl6 36	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ¬ 𝐴 ≈ 𝐵))
26	9, 25	sylbird 263	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≠ 𝐵 → ¬ 𝐴 ≈ 𝐵))
27	26	necon4ad 2976	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ≈ 𝐵 → ran 𝐹 = 𝐵))
28	5, 27	mpd 16	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵)
29		df-fo 6543	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
30	4, 28, 29	sylanbrc 595	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–onto→𝐵)
31		df-f1o 6544	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
32	1, 30, 31	sylanbrc 595	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
33	32	ex 418	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵))
34		f1of1 6820	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
35	33, 34	impbid1 228	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ⊆ wss 3902   ⊊ wpss 3903   class class class wbr 5107  ran crn 5660   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536   ≈ cen 8953   ≼ cdom 8954   ≺ csdm 8955  Fincfn 8956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7867  df-1o 8459  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960

This theorem is used by:  hashfac  14527  crth  16875  eulerthlem2  16879  fidomndrnglem  20945  mdetunilem8  22847  basellem4  27328  lgsqrlem4  27593  lgseisenlem2  27620  aks5lem7  43074