Theorem fidomdm 9402

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 6231	. 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹
2		finresfin 9332	. . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3		fvex 6933	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
4		eqid 2740	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))
5	3, 4	fnmpti 6723	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V)
6		dffn4 6840	. . . . . 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 230	. . . . 5 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))
8		relres 6035	. . . . . 6 ⊢ Rel (𝐹 ↾ V)
9		reldm 8085	. . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))
10		foeq3 6832	. . . . . 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 231	. . . 4 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13		fodomfi 9378	. . . 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 585	. . 3 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15		resss 6031	. . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹
16		ssdomg 9060	. . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
17	15, 16	mpi 20	. . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18		domtr 9067	. . 3 ⊢ ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
19	14, 17, 18	syl2anc 583	. 2 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
20	1, 19	eqbrtrrid 5202	1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ↾ cres 5702  Rel wrel 5705   Fn wfn 6568  –onto→wfo 6571  ‘cfv 6573  1^st c1st 8028   ≼ cdom 9001  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-en 9004  df-dom 9005  df-fin 9007

This theorem is referenced by:  dmfi  9403  hashfun  14486