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Theorem dmct 9938
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 6054 . 2 dom (𝐴 ↾ V) = dom 𝐴
2 resss 5876 . . . . 5 (𝐴 ↾ V) ⊆ 𝐴
3 ctex 8516 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
4 ssexg 5223 . . . . 5 (((𝐴 ↾ V) ⊆ 𝐴𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
52, 3, 4sylancr 587 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6 fvex 6679 . . . . . . 7 (1st𝑥) ∈ V
7 eqid 2825 . . . . . . 7 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
86, 7fnmpti 6487 . . . . . 6 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V)
9 dffn4 6592 . . . . . 6 ((𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V) ↔ (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
108, 9mpbi 231 . . . . 5 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
11 relres 5880 . . . . . 6 Rel (𝐴 ↾ V)
12 reldm 7737 . . . . . 6 (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
13 foeq3 6584 . . . . . 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 232 . . . 4 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)):(𝐴 ↾ V)–onto→dom (𝐴 ↾ V)
16 fodomg 9937 . . . 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 8547 . . . . 5 (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
193, 2, 18mpisyl 21 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20 domtr 8554 . . . 4 (((𝐴 ↾ V) ≼ 𝐴𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
2119, 20mpancom 684 . . 3 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22 domtr 8554 . . 3 ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
2317, 21, 22syl2anc 584 . 2 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
241, 23eqbrtrrid 5098 1 (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  Vcvv 3499  wss 3939   class class class wbr 5062  cmpt 5142  dom cdm 5553  ran crn 5554  cres 5555  Rel wrel 5558   Fn wfn 6346  ontowfo 6349  cfv 6351  ωcom 7571  1st c1st 7681  cdom 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-ac2 9877
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  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 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-card 9360  df-acn 9363  df-ac 9534
This theorem is referenced by:  rnct  9939
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