Theorem f1finf1o 9211

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 5319. (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 488	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵)
2		f1f 6755	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	adantl 485	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴⟶𝐵)
4	3	ffnd 6687	. . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
5		simpll 776	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵)
6	3	frnd 6695	. . . . . . . . 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 543	. . . . . . . . 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 9171	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊊ 𝐵) → ran 𝐹 ≺ 𝐵)
11	10	ex 416	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
12	11	ad2antlr 737	. . . . . . . . . 10 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ⊊ 𝐵 → ran 𝐹 ≺ 𝐵))
13		enfii 9148	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
14	13	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
15		f1f1orn 6813	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
16		f1oenfi 9141	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→ran 𝐹) → 𝐴 ≈ ran 𝐹)
17	14, 15, 16	syl2an 605	. . . . . . . . . . 11 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ ran 𝐹)
18		endom 8954	. . . . . . . . . . . . 13 ⊢ (𝐴 ≈ ran 𝐹 → 𝐴 ≼ ran 𝐹)
19		domsdomtrfi 9164	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
20	18, 19	syl3an2 1176	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹 ∧ ran 𝐹 ≺ 𝐵) → 𝐴 ≺ 𝐵)
21	20	3expia 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ ran 𝐹) → (ran 𝐹 ≺ 𝐵 → 𝐴 ≺ 𝐵))
22	14, 17, 21	syl2an2r 695	. . . . . . . . . 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 8956	. . . . . . . . 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 262	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (ran 𝐹 ≠ 𝐵 → ¬ 𝐴 ≈ 𝐵))
27	26	necon4ad 2975	. . . . . 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 6522	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
30	4, 28, 29	sylanbrc 592	. . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–onto→𝐵)
31		df-f1o 6523	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵))
32	1, 30, 31	sylanbrc 592	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
33	32	ex 416	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵))
34		f1of1 6800	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
35	33, 34	impbid1 227	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ⊆ wss 3902   ⊊ wpss 3903   class class class wbr 5097  ran crn 5644   Fn wfn 6511  ⟶wf 6512  –1-1→wf1 6513  –onto→wfo 6514  –1-1-onto→wf1o 6515   ≈ cen 8918   ≼ cdom 8919   ≺ csdm 8920  Fincfn 8921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  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 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-om 7842  df-1o 8431  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925

This theorem is referenced by:  hashfac  14465  crth  16804  eulerthlem2  16808  fidomndrnglem  20809  mdetunilem8  22667  basellem4  27136  lgsqrlem4  27401  lgseisenlem2  27428  aks5lem7  42778