Theorem f1finf1o 9302

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⟶1-1𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))

Proof of Theorem f1finf1o

Step	Hyp	Ref	Expression
1		simpr 490	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹:𝐴⟶1-1𝐵)
2		f1f 6808	. . . . . . 7 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶𝐵)
3	2	adantl 487	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹:𝐴⟶𝐵)
4	3	ffnd 6740	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹 Fn 𝐴)
5		simpll 779	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≈ 𝐵)
6	3	frnd 6748	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → ran 𝐹 ⊆ 𝐵)
7		df-pss 4017	. . . . . . . . . 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⟶1-1𝐵) → (ran 𝐹 ⊊ 𝐵 ↔ ran 𝐹 ≠ 𝐵))
10		php3 9262	. . . . . . . . . . . 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⟶1-1𝐵) → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
13		enfii 9239	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
14	13	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
15		f1f1orn 6866	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶1-1ontoran 𝐹)
16		f1oenfi 9232	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶1-1ontoran 𝐹) → 𝐴 ≈ ran 𝐹)
17	14, 15, 16	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≈ ran 𝐹)
18		endom 9044	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ ran 𝐹 → 𝐴 ≼ ran 𝐹)
19		domsdomtrfi 9255	. . . . . . . . . . . . 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⟶1-1𝐵) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
23	12, 22	syld 48	. . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → (ran 𝐹 ⊊ 𝐵 → 𝐴 ≺ 𝐵))
24		sdomnen 9046	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
25	23, 24	syl6 36	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → (ran 𝐹 ⊊ 𝐵 → ¬ 𝐴 ≈ 𝐵))
26	9, 25	sylbird 263	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → (ran 𝐹 ≠ 𝐵 → ¬ 𝐴 ≈ 𝐵))
27	26	necon4ad 2978	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → (𝐴 ≈ 𝐵 → ran 𝐹 = 𝐵))
28	5, 27	mpd 16	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → ran 𝐹 = 𝐵)
29		df-fo 6572	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
30	4, 28, 29	sylanbrc 595	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹:𝐴⟶onto𝐵)
31		df-f1o 6573	. . . 4 ⊢ (𝐹:𝐴⟶1-1onto𝐵 ↔ (𝐹:𝐴⟶1-1𝐵 ∧ 𝐹:𝐴⟶onto𝐵))
32	1, 30, 31	sylanbrc 595	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹:𝐴⟶1-1onto𝐵)
33	32	ex 418	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶1-1onto𝐵))
34		f1of1 6853	. 2 ⊢ (𝐹:𝐴⟶1-1onto𝐵 → 𝐹:𝐴⟶1-1𝐵)
35	33, 34	impbid1 228	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴⟶1-1𝐵 ↔ 𝐹:𝐴⟶1-1onto𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ⊊ wpss 4016   class class class wbr 5108  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010  Fincfn 9011

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1o 8514  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015

This theorem is used by:  hashfac  14644  crth  16999  eulerthlem2  17003  fidomndrnglem  21081  mdetunilem8  22989  basellem4  27468  lgsqrlem4  27733  lgseisenlem2  27760  aks5lem7  43516