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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 6220 | . 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹 | |
| 2 | finresfin 9304 | . . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin) | |
| 3 | fvex 6919 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
| 4 | eqid 2737 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) | |
| 5 | 3, 4 | fnmpti 6711 | . . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) |
| 6 | dffn4 6826 | . . . . . 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 6023 | . . . . . 6 ⊢ Rel (𝐹 ↾ V) | |
| 9 | reldm 8069 | . . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) | |
| 10 | foeq3 6818 | . . . . . 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 9350 | . . . 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 6019 | . . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹 | |
| 16 | ssdomg 9040 | . . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹)) | |
| 17 | 15, 16 | mpi 20 | . . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹) |
| 18 | domtr 9047 | . . 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 5179 | 1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ↾ cres 5687 Rel wrel 5690 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 1st c1st 8012 ≼ cdom 8983 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: dmfi 9375 hashfun 14476 |
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