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Theorem dmct 9291
Description: The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
dmct (𝐴 ≼ ω → dom 𝐴 ≼ ω)

Proof of Theorem dmct
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmresv 5555 . 2 dom (𝐴 ↾ V) = dom 𝐴
2 resss 5385 . . . . 5 (𝐴 ↾ V) ⊆ 𝐴
3 ctex 7915 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
4 ssexg 4769 . . . . 5 (((𝐴 ↾ V) ⊆ 𝐴𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
52, 3, 4sylancr 694 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6 fvex 6160 . . . . . . 7 (1st𝑥) ∈ V
7 eqid 2626 . . . . . . 7 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
86, 7fnmpti 5981 . . . . . 6 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V)
9 dffn4 6080 . . . . . 6 ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
108, 9mpbi 220 . . . . 5 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
11 relres 5389 . . . . . 6 Rel (𝐴 ↾ V)
12 reldm 7167 . . . . . 6 (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
13 foeq3 6072 . . . . . 6 (dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))))
1411, 12, 13mp2b 10 . . . . 5 ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
1510, 14mpbir 221 . . . 4 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
16 fodomg 9290 . . . 4 ((𝐴 ↾ V) ∈ V → ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V) → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V)))
175, 15, 16mpisyl 21 . . 3 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ (𝐴 ↾ V))
18 ssdomg 7946 . . . . 5 (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
193, 2, 18mpisyl 21 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20 domtr 7954 . . . 4 (((𝐴 ↾ V) ≼ 𝐴𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
2119, 20mpancom 702 . . 3 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22 domtr 7954 . . 3 ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
2317, 21, 22syl2anc 692 . 2 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
241, 23syl5eqbrr 4654 1 (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1992  Vcvv 3191  wss 3560   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cres 5081  Rel wrel 5084   Fn wfn 5845  ontowfo 5848  cfv 5850  ωcom 7013  1st c1st 7114  cdom 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-ac2 9230
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-card 8710  df-acn 8713  df-ac 8884
This theorem is referenced by:  rnct  9292
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