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Theorem fnct 9957
Description: If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
fnct ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)

Proof of Theorem fnct
StepHypRef Expression
1 ctex 8520 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
21adantl 485 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐴 ∈ V)
3 fndm 6443 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43eleq1d 2900 . . . . . . 7 (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
54adantr 484 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
62, 5mpbird 260 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → dom 𝐹 ∈ V)
7 fnfun 6441 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
87adantr 484 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → Fun 𝐹)
9 funrnex 7650 . . . . 5 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
106, 8, 9sylc 65 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ∈ V)
112, 10xpexd 7468 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12 simpl 486 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 Fn 𝐴)
13 dffn3 6515 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1412, 13sylib 221 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
15 fssxp 6524 . . . 4 (𝐹:𝐴⟶ran 𝐹𝐹 ⊆ (𝐴 × ran 𝐹))
1614, 15syl 17 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
17 ssdomg 8551 . . 3 ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
1811, 16, 17sylc 65 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
19 xpdom1g 8610 . . . . 5 ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
2010, 19sylancom 591 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
21 omex 9103 . . . . 5 ω ∈ V
22 fnrndomg 9956 . . . . . . 7 (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
232, 12, 22sylc 65 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹𝐴)
24 domtr 8558 . . . . . 6 ((ran 𝐹𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
2523, 24sylancom 591 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26 xpdom2g 8609 . . . . 5 ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
2721, 25, 26sylancr 590 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
28 domtr 8558 . . . 4 (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
2920, 27, 28syl2anc 587 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
30 xpomen 9439 . . 3 (ω × ω) ≈ ω
31 domentr 8564 . . 3 (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
3229, 30, 31sylancl 589 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
33 domtr 8558 . 2 ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
3418, 32, 33syl2anc 587 1 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  Vcvv 3480  wss 3919   class class class wbr 5052   × cxp 5540  dom cdm 5542  ran crn 5543  Fun wfun 6337   Fn wfn 6338  wf 6339  ωcom 7574  cen 8502  cdom 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-ac2 9883
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-oi 8971  df-card 9365  df-acn 9368  df-ac 9540
This theorem is referenced by:  mptct  9958  mpocti  30465  mptctf  30467  omssubadd  31618
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