Theorem fodomfi 9212

Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodomg 10432 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 5310. (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 6751	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) = 𝐵)
3		imaeq2 6015	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
4		ima0 6036	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
5	3, 4	eqtrdi 2787	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = ∅)
6		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
7	5, 6	breq12d 5111	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9		imaeq2 6015	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	breq12d 5111	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝑦) ≼ 𝑦))
12	11	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦)))
13		imaeq2 6015	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14		id 22	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
15	13, 14	breq12d 5111	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
16	15	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
17		imaeq2 6015	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 “ 𝑥) = (𝐹 “ 𝐴))
18		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
19	17, 18	breq12d 5111	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝐴) ≼ 𝐴))
20	19	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴)))
21		0ex 5252	. . . . . 6 ⊢ ∅ ∈ V
22	21	0dom 9035	. . . . 5 ⊢ ∅ ≼ ∅
23	22	a1i 11	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24		fnfun 6592	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
25	24	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → Fun 𝐹)
26		funressn 7104	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
27		rnss 5888	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩} → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
28	25, 26, 27	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
29		df-ima 5637	. . . . . . . . . . . 12 ⊢ (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
31	30	rnsnop 6182	. . . . . . . . . . . . 13 ⊢ ran {⟨𝑧, (𝐹‘𝑧)⟩} = {(𝐹‘𝑧)}
32	31	eqcomi 2745	. . . . . . . . . . . 12 ⊢ {(𝐹‘𝑧)} = ran {⟨𝑧, (𝐹‘𝑧)⟩}
33	28, 29, 32	3sstr4g 3987	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)})
34		snfi 8980	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ∈ Fin
35		ssexg 5268	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ∈ Fin) → (𝐹 “ {𝑧}) ∈ V)
36	33, 34, 35	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37		fvi 6910	. . . . . . . . . 10 ⊢ ((𝐹 “ {𝑧}) ∈ V → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
39	38	uneq2d 4120	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})))
40		imaundi 6107	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
41	39, 40	eqtr4di 2789	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
42		simprr 772	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ 𝑦) ≼ 𝑦)
43		ssdomfi 9120	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ∈ Fin → ((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)}))
44	34, 33, 43	mpsyl 68	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)})
45		fvex 6847	. . . . . . . . . . 11 ⊢ (𝐹‘𝑧) ∈ V
46		en2sn 8978	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑧) ∈ V ∧ 𝑧 ∈ V) → {(𝐹‘𝑧)} ≈ {𝑧})
47	45, 30, 46	mp2an 692	. . . . . . . . . 10 ⊢ {(𝐹‘𝑧)} ≈ {𝑧}
48		endom 8916	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ≈ {𝑧} → {(𝐹‘𝑧)} ≼ {𝑧})
49		domtrfi 9117	. . . . . . . . . . . 12 ⊢ (({(𝐹‘𝑧)} ∈ Fin ∧ (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
50	34, 49	mp3an1 1450	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
51	48, 50	sylan2 593	. . . . . . . . . 10 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
52	44, 47, 51	sylancl 586	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
53	38, 52	eqbrtrd 5120	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
54		simplr 768	. . . . . . . . 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 234	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
57		undom 8993	. . . . . . . 8 ⊢ ((((𝐹 “ 𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
58	42, 53, 56, 57	syl21anc 837	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
59	41, 58	eqbrtrrd 5122	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
60	59	exp32 420	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹 Fn 𝐴 → ((𝐹 “ 𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
61	60	a2d 29	. . . 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 9090	. . 3 ⊢ (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴))
63		fofn 6748	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
64	62, 63	impel 505	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) ≼ 𝐴)
65	2, 64	eqbrtrrd 5122	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586   class class class wbr 5098   I cid 5518  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-dom 8885  df-fin 8887

This theorem is referenced by:  fofi  9213  fodomfib  9229  fodomfibOLD  9231  fofinf1o  9232  fidomdm  9234  cmpsub  23344  alexsubALT  23995  phpreu  37805  poimirlem26  37847  imadomfi  42266