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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 6060 | . 2 ⊢ dom (𝐴 ↾ V) = dom 𝐴 | |
2 | resss 5881 | . . . . 5 ⊢ (𝐴 ↾ V) ⊆ 𝐴 | |
3 | ctex 8527 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
4 | ssexg 5230 | . . . . 5 ⊢ (((𝐴 ↾ V) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝐴 ↾ V) ∈ V) | |
5 | 2, 3, 4 | sylancr 589 | . . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V) |
6 | fvex 6686 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
7 | eqid 2824 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) | |
8 | 6, 7 | fnmpti 6494 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐴 ↾ V) |
9 | dffn4 6599 | . . . . . 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 232 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)) |
11 | relres 5885 | . . . . . 6 ⊢ Rel (𝐴 ↾ V) | |
12 | reldm 7746 | . . . . . 6 ⊢ (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥))) | |
13 | foeq3 6591 | . . . . . 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 233 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st ‘𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) |
16 | fodomg 9948 | . . . 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 8558 | . . . . 5 ⊢ (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴)) | |
19 | 3, 2, 18 | mpisyl 21 | . . . 4 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴) |
20 | domtr 8565 | . . . 4 ⊢ (((𝐴 ↾ V) ≼ 𝐴 ∧ 𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω) | |
21 | 19, 20 | mpancom 686 | . . 3 ⊢ (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω) |
22 | domtr 8565 | . . 3 ⊢ ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω) | |
23 | 17, 21, 22 | syl2anc 586 | . 2 ⊢ (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω) |
24 | 1, 23 | eqbrtrrid 5105 | 1 ⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 class class class wbr 5069 ↦ cmpt 5149 dom cdm 5558 ran crn 5559 ↾ cres 5560 Rel wrel 5563 Fn wfn 6353 –onto→wfo 6356 ‘cfv 6358 ωcom 7583 1st c1st 7690 ≼ cdom 8510 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-ac2 9888 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-card 9371 df-acn 9374 df-ac 9545 |
This theorem is referenced by: rnct 9950 |
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