Theorem fodomfi 9269

Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 10459 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)

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 6761	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	adantl 482	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) = 𝐵)
3		imaeq2 6009	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
4		ima0 6029	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
5	3, 4	eqtrdi 2792	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = ∅)
6		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
7	5, 6	breq12d 5118	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9		imaeq2 6009	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	breq12d 5118	. . . . 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 6009	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14		id 22	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
15	13, 14	breq12d 5118	. . . . 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 6009	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 “ 𝑥) = (𝐹 “ 𝐴))
18		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
19	17, 18	breq12d 5118	. . . . 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 5264	. . . . . 6 ⊢ ∅ ∈ V
22	21	0dom 9050	. . . . 5 ⊢ ∅ ≼ ∅
23	22	a1i 11	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24		fnfun 6602	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
25	24	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → Fun 𝐹)
26		funressn 7105	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
27		rnss 5894	. . . . . . . . . . . . 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 5646	. . . . . . . . . . . 12 ⊢ (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30		vex 3449	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
31	30	rnsnop 6176	. . . . . . . . . . . . 13 ⊢ ran {⟨𝑧, (𝐹‘𝑧)⟩} = {(𝐹‘𝑧)}
32	31	eqcomi 2745	. . . . . . . . . . . 12 ⊢ {(𝐹‘𝑧)} = ran {⟨𝑧, (𝐹‘𝑧)⟩}
33	28, 29, 32	3sstr4g 3989	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)})
34		snex 5388	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ∈ V
35		ssexg 5280	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ∈ V) → (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 6917	. . . . . . . . . 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 4123	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})))
40		imaundi 6102	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
41	39, 40	eqtr4di 2794	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
42		simprr 771	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ 𝑦) ≼ 𝑦)
43		ssdomg 8940	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ∈ V → ((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 6855	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑧) ∈ V
46	45	ensn1 8961	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ≈ 1o
47	30	ensn1 8961	. . . . . . . . . . 11 ⊢ {𝑧} ≈ 1o
48	46, 47	entr4i 8951	. . . . . . . . . 10 ⊢ {(𝐹‘𝑧)} ≈ {𝑧}
49		domentr 8953	. . . . . . . . . 10 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
50	44, 48, 49	sylancl 586	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
51	38, 50	eqbrtrd 5127	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
52		simplr 767	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
53		disjsn 4672	. . . . . . . . 9 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
54	52, 53	sylibr 233	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
55		undom 9003	. . . . . . . 8 ⊢ ((((𝐹 “ 𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
56	42, 51, 54, 55	syl21anc 836	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
57	41, 56	eqbrtrrd 5129	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
58	57	exp32 421	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹 Fn 𝐴 → ((𝐹 “ 𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
59	58	a2d 29	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
60	8, 12, 16, 20, 23, 59	findcard2s 9109	. . 3 ⊢ (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴))
61		fofn 6758	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
62	60, 61	impel 506	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) ≼ 𝐴)
63	2, 62	eqbrtrrd 5129	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586  ⟨cop 4592   class class class wbr 5105   I cid 5530  ran crn 5634   ↾ cres 5635   “ cima 5636  Fun wfun 6490   Fn wfn 6491  –onto→wfo 6494  ‘cfv 6496  1_oc1o 8405   ≈ 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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887

This theorem is referenced by:  fodomfib  9270  fofinf1o  9271  fidomdm  9273  fofi  9282  pwfilemOLD  9290  cmpsub  22751  alexsubALT  23402  phpreu  36062  poimirlem26  36104