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Mirrors > Home > MPE Home > Th. List > fidomdm | Structured version Visualization version GIF version |
Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
fidomdm | ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmresv 6199 | . 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹 | |
2 | finresfin 9276 | . . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin) | |
3 | fvex 6904 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
4 | eqid 2731 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) | |
5 | 3, 4 | fnmpti 6693 | . . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) |
6 | dffn4 6811 | . . . . . 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 229 | . . . . 5 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) |
8 | relres 6010 | . . . . . 6 ⊢ Rel (𝐹 ↾ V) | |
9 | reldm 8034 | . . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) | |
10 | foeq3 6803 | . . . . . 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 230 | . . . 4 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) |
13 | fodomfi 9331 | . . . 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 6006 | . . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹 | |
16 | ssdomg 9002 | . . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹)) | |
17 | 15, 16 | mpi 20 | . . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹) |
18 | domtr 9009 | . . 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 5184 | 1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 Rel wrel 5681 Fn wfn 6538 –onto→wfo 6541 ‘cfv 6543 1st c1st 7977 ≼ cdom 8943 Fincfn 8945 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7860 df-1st 7979 df-2nd 7980 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-fin 8949 |
This theorem is referenced by: dmfi 9336 hashfun 14404 |
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