Theorem fidomdm 9279

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 6190	. 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹
2		finresfin 9220	. . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3		fvex 6884	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
4		eqid 2765	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))
5	3, 4	fnmpti 6668	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V)
6		dffn4 6788	. . . . . 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 5994	. . . . . 6 ⊢ Rel (𝐹 ↾ V)
9		reldm 8029	. . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))
10		foeq3 6780	. . . . . 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 9260	. . . 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 597	. . 3 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15		resss 5990	. . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹
16		ssdomg 8985	. . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
17	15, 16	mpi 21	. . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18		domtr 8992	. . 3 ⊢ ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
19	14, 17, 18	syl2anc 595	. 2 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
20	1, 19	eqbrtrrid 5140	1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907   class class class wbr 5104   ↦ cmpt 5185  dom cdm 5651  ran crn 5652   ↾ cres 5653  Rel wrel 5656   Fn wfn 6520  –onto→wfo 6523  ‘cfv 6525  1^st c1st 7972   ≼ cdom 8929  Fincfn 8931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  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 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-om 7851  df-1st 7974  df-2nd 7975  df-1o 8441  df-en 8932  df-dom 8933  df-fin 8935

This theorem is referenced by:  dmfi  9280  hashfun  14462