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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 6222 | . 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹 | |
2 | finresfin 9302 | . . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin) | |
3 | fvex 6920 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
4 | eqid 2735 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) | |
5 | 3, 4 | fnmpti 6712 | . . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) |
6 | dffn4 6827 | . . . . . 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 6026 | . . . . . 6 ⊢ Rel (𝐹 ↾ V) | |
9 | reldm 8068 | . . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) | |
10 | foeq3 6819 | . . . . . 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 9348 | . . . 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 586 | . . 3 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V)) |
15 | resss 6022 | . . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹 | |
16 | ssdomg 9039 | . . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹)) | |
17 | 15, 16 | mpi 20 | . . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹) |
18 | domtr 9046 | . . 3 ⊢ ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹) | |
19 | 14, 17, 18 | syl2anc 584 | . 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 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ↾ cres 5691 Rel wrel 5694 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 1st c1st 8011 ≼ cdom 8982 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-en 8985 df-dom 8986 df-fin 8988 |
This theorem is referenced by: dmfi 9373 hashfun 14473 |
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