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Theorem dmct 28877
 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 5511 . 2 dom (𝐴 ↾ V) = dom 𝐴
2 resss 5342 . . . . 5 (𝐴 ↾ V) ⊆ 𝐴
3 ctex 7856 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
4 ssexg 4732 . . . . 5 (((𝐴 ↾ V) ⊆ 𝐴𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
52, 3, 4sylancr 694 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6 fvex 6113 . . . . . . 7 (1st𝑥) ∈ V
7 eqid 2610 . . . . . . 7 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
86, 7fnmpti 5935 . . . . . 6 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V)
9 dffn4 6034 . . . . . 6 ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
108, 9mpbi 219 . . . . 5 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
11 relres 5346 . . . . . 6 Rel (𝐴 ↾ V)
12 reldm 7110 . . . . . 6 (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
13 foeq3 6026 . . . . . 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 220 . . . 4 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
16 fodomg 9228 . . . 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 7887 . . . . 5 (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
193, 2, 18mpisyl 21 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20 domtr 7895 . . . 4 (((𝐴 ↾ V) ≼ 𝐴𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
2119, 20mpancom 700 . . 3 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22 domtr 7895 . . 3 ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
2317, 21, 22syl2anc 691 . 2 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
241, 23syl5eqbrr 4619 1 (𝐴 ≼ ω → dom 𝐴 ≼ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  ωcom 6957  1st c1st 7057   ≼ cdom 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-card 8648  df-acn 8651  df-ac 8822 This theorem is referenced by:  rnct  28879
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