Theorem fidomdm 9316

Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
fidomdm	⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)

Proof of Theorem fidomdm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmresv 6193	. 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹
2		finresfin 9256	. . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3		fvex 6896	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
4		eqid 2761	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))
5	3, 4	fnmpti 6680	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V)
6		dffn4 6800	. . . . . 6 ⊢ ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))
7	5, 6	mpbi 233	. . . . 5 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))
8		relres 5996	. . . . . 6 ⊢ Rel (𝐹 ↾ V)
9		reldm 8053	. . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))
10		foeq3 6792	. . . . . 6 ⊢ (dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) → ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))))
11	8, 9, 10	mp2b 10	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))
12	7, 11	mpbir 234	. . . 4 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13		fodomfi 9297	. . . 4 ⊢ (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
14	2, 12, 13	sylancl 598	. . 3 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15		resss 5992	. . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹
16		ssdomg 9020	. . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
17	15, 16	mpi 21	. . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18		domtr 9027	. . 3 ⊢ ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
19	14, 17, 18	syl2anc 596	. 2 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
20	1, 19	eqbrtrrid 5141	1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652   ↾ cres 5653  Rel wrel 5656   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  1^st c1st 7997   ≼ cdom 8964  Fincfn 8966

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7876  df-1st 7999  df-2nd 8000  df-1o 8469  df-en 8967  df-dom 8968  df-fin 8970

This theorem is used by:  dmfi  9317  hashfun  14575