Theorem fodomfi 9312

Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodomg 10607 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5327. (Revised by BTernaryTau, 20-Jun-2025.)

Assertion

Ref	Expression
fodomfi	⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Proof of Theorem fodomfi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foima 6806	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	adantl 487	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) = 𝐵)
3		imaeq2 6050	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
4		ima0 6077	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
5	3, 4	eqtrdi 2811	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = ∅)
6		id 23	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
7	5, 6	breq12d 5115	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9		imaeq2 6050	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	breq12d 5115	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝑦) ≼ 𝑦))
12	11	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦)))
13		imaeq2 6050	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14		id 23	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
15	13, 14	breq12d 5115	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
16	15	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
17		imaeq2 6050	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 “ 𝑥) = (𝐹 “ 𝐴))
18		id 23	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
19	17, 18	breq12d 5115	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝐴) ≼ 𝐴))
20	19	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴)))
21		0ex 5260	. . . . . 6 ⊢ ∅ ∈ V
22	21	0dom 9134	. . . . 5 ⊢ ∅ ≼ ∅
23	22	a1i 11	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24		fnfun 6644	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
25	24	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → Fun 𝐹)
26		funressn 7168	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
27		rnss 5922	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩} → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
28	25, 26, 27	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
29		df-ima 5664	. . . . . . . . . . . 12 ⊢ (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
31	30	rnsnop 6228	. . . . . . . . . . . . 13 ⊢ ran {⟨𝑧, (𝐹‘𝑧)⟩} = {(𝐹‘𝑧)}
32	31	eqcomi 2769	. . . . . . . . . . . 12 ⊢ {(𝐹‘𝑧)} = ran {⟨𝑧, (𝐹‘𝑧)⟩}
33	28, 29, 32	3sstr4g 3962	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)})
34		snfi 9079	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ∈ Fin
35		ssexg 5281	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ∈ Fin) → (𝐹 “ {𝑧}) ∈ V)
36	33, 34, 35	sylancl 598	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37		fvi 6966	. . . . . . . . . 10 ⊢ ((𝐹 “ {𝑧}) ∈ V → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
38	36, 37	syl 18	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
39	38	uneq2d 4166	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})))
40		imaundi 6143	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
41	39, 40	eqtr4di 2813	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
42		simprr 785	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ 𝑦) ≼ 𝑦)
43		ssdomfi 9219	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ∈ Fin → ((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)}))
44	34, 33, 43	mpsyl 69	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)})
45		fvex 6903	. . . . . . . . . . 11 ⊢ (𝐹‘𝑧) ∈ V
46		en2sn 9077	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑧) ∈ V ∧ 𝑧 ∈ V) → {(𝐹‘𝑧)} ≈ {𝑧})
47	45, 30, 46	mp2an 705	. . . . . . . . . 10 ⊢ {(𝐹‘𝑧)} ≈ {𝑧}
48		endom 9014	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ≈ {𝑧} → {(𝐹‘𝑧)} ≼ {𝑧})
49		domtrfi 9216	. . . . . . . . . . . 12 ⊢ (({(𝐹‘𝑧)} ∈ Fin ∧ (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
50	34, 49	mp3an1 1477	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
51	48, 50	sylan2 605	. . . . . . . . . 10 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
52	44, 47, 51	sylancl 598	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
53	38, 52	eqbrtrd 5126	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
54		simplr 781	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
55		disjsn 4668	. . . . . . . . 9 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
56	54, 55	sylibr 237	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
57		undom 9092	. . . . . . . 8 ⊢ ((((𝐹 “ 𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
58	42, 53, 56, 57	syl21anc 851	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
59	41, 58	eqbrtrrd 5128	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
60	59	exp32 426	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹 Fn 𝐴 → ((𝐹 “ 𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
61	60	a2d 30	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
62	8, 12, 16, 20, 23, 61	findcard2s 9189	. . 3 ⊢ (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴))
63		fofn 6803	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
64	62, 63	impel 515	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) ≼ 𝐴)
65	2, 64	eqbrtrrd 5128	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  {csn 4584  ⟨cop 4834   class class class wbr 5102   I cid 5545  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6537   Fn wfn 6538  –onto→wfo 6541  ‘cfv 6543   ≈ cen 8978   ≼ cdom 8979  Fincfn 8981

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7882  df-1o 8484  df-en 8982  df-dom 8983  df-fin 8985

This theorem is used by:  fofi  9313  fodomfib  9328  fofinf1o  9329  fidomdm  9331  cmpsub  23727  alexsubALT  24379  phpreu  38664  poimirlem26  38701  imadomfi  43189  disjinfi  46343