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Theorem dmct 9548
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 5734 . 2 dom (𝐴 ↾ V) = dom 𝐴
2 resss 5563 . . . . 5 (𝐴 ↾ V) ⊆ 𝐴
3 ctex 8124 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
4 ssexg 4938 . . . . 5 (((𝐴 ↾ V) ⊆ 𝐴𝐴 ∈ V) → (𝐴 ↾ V) ∈ V)
52, 3, 4sylancr 575 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ∈ V)
6 fvex 6342 . . . . . . 7 (1st𝑥) ∈ V
7 eqid 2771 . . . . . . 7 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥))
86, 7fnmpti 6162 . . . . . 6 (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)) Fn (𝐴 ↾ V)
9 dffn4 6262 . . . . . 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 5567 . . . . . 6 Rel (𝐴 ↾ V)
12 reldm 7368 . . . . . 6 (Rel (𝐴 ↾ V) → dom (𝐴 ↾ V) = ran (𝑥 ∈ (𝐴 ↾ V) ↦ (1st𝑥)))
13 foeq3 6254 . . . . . 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 9547 . . . 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 8155 . . . . 5 (𝐴 ∈ V → ((𝐴 ↾ V) ⊆ 𝐴 → (𝐴 ↾ V) ≼ 𝐴))
193, 2, 18mpisyl 21 . . . 4 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ 𝐴)
20 domtr 8162 . . . 4 (((𝐴 ↾ V) ≼ 𝐴𝐴 ≼ ω) → (𝐴 ↾ V) ≼ ω)
2119, 20mpancom 668 . . 3 (𝐴 ≼ ω → (𝐴 ↾ V) ≼ ω)
22 domtr 8162 . . 3 ((dom (𝐴 ↾ V) ≼ (𝐴 ↾ V) ∧ (𝐴 ↾ V) ≼ ω) → dom (𝐴 ↾ V) ≼ ω)
2317, 21, 22syl2anc 573 . 2 (𝐴 ≼ ω → dom (𝐴 ↾ V) ≼ ω)
241, 23syl5eqbrr 4822 1 (𝐴 ≼ ω → dom 𝐴 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723   class class class wbr 4786  cmpt 4863  dom cdm 5249  ran crn 5250  cres 5251  Rel wrel 5254   Fn wfn 6026  ontowfo 6029  cfv 6031  ωcom 7212  1st c1st 7313  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-ac2 9487
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-card 8965  df-acn 8968  df-ac 9139
This theorem is referenced by:  rnct  9549
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