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Mirrors > Home > MPE Home > Th. List > dmct | Structured version Visualization version GIF version |
Description: The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
dmct | ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmresv 6024 | . 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴 | |
2 | resss 5843 | . . . . 5 ⊢ (𝐴 ↾ V) ⊆ 𝐴 | |
3 | ctex 8507 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
4 | ssexg 5191 | . . . . 5 ⊢ (((𝐴 ↾ V) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ↾ V) ∈ V) | |
5 | 2, 3, 4 | sylancr 590 | . . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V) |
6 | fvex 6658 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
7 | eqid 2798 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) | |
8 | 6, 7 | fnmpti 6463 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) |
9 | dffn4 6571 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))) | |
10 | 8, 9 | mpbi 233 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) |
11 | relres 5847 | . . . . . 6 ⊢ Rel (𝐴 ↾ V) | |
12 | reldm 7725 | . . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))) | |
13 | foeq3 6563 | . . . . . 6 ⊢ (dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)))) | |
14 | 11, 12, 13 | mp2b 10 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))) |
15 | 10, 14 | mpbir 234 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) |
16 | fodomg 9933 | . . . 4 ⊢ ((𝐴 ↾ V) ∈ V → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V))) | |
17 | 5, 15, 16 | mpisyl 21 | . . 3 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V)) |
18 | ssdomg 8538 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴)) | |
19 | 3, 2, 18 | mpisyl 21 | . . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴) |
20 | domtr 8545 | . . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω) | |
21 | 19, 20 | mpancom 687 | . . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω) |
22 | domtr 8545 | . . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω) | |
23 | 17, 21, 22 | syl2anc 587 | . 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω) |
24 | 1, 23 | eqbrtrrid 5066 | 1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ↾ cres 5521 Rel wrel 5524 Fn wfn 6319 –onto→wfo 6322 ‘cfv 6324 ωcom 7560 1st c1st 7669 ≼ cdom 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-card 9352 df-acn 9355 df-ac 9527 |
This theorem is referenced by: rnct 9936 |
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