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Theorem dmct 9459
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 5703 . 2 dom (𝐴 ↾ V) = dom 𝐴
2 resss 5532 . . . . 5 (𝐴 ↾ V) ⊆ 𝐴
3 ctex 8087 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
4 ssexg 4912 . . . . 5 (((𝐴 ↾ V) ⊆ 𝐴𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
52, 3, 4sylancr 698 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6 fvex 6314 . . . . . . 7 (1st𝑥) ∈ V
7 eqid 2724 . . . . . . 7 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
86, 7fnmpti 6135 . . . . . 6 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V)
9 dffn4 6234 . . . . . 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 5536 . . . . . 6 Rel (𝐴 ↾ V)
12 reldm 7338 . . . . . 6 (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
13 foeq3 6226 . . . . . 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 9458 . . . 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 8118 . . . . 5 (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
193, 2, 18mpisyl 21 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20 domtr 8125 . . . 4 (((𝐴 ↾ V) ≼ 𝐴𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
2119, 20mpancom 706 . . 3 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22 domtr 8125 . . 3 ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
2317, 21, 22syl2anc 696 . 2 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
241, 23syl5eqbrr 4796 1 (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1596  wcel 2103  Vcvv 3304  wss 3680   class class class wbr 4760  cmpt 4837  dom cdm 5218  ran crn 5219  cres 5220  Rel wrel 5223   Fn wfn 5996  ontowfo 5999  cfv 6001  ωcom 7182  1st c1st 7283  cdom 8070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-ac2 9398
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-card 8878  df-acn 8881  df-ac 9052
This theorem is referenced by:  rnct  9460
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