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Theorem fnct 9947
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 8512 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
21adantl 482 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐴 ∈ V)
3 fndm 6448 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43eleq1d 2894 . . . . . . 7 (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
54adantr 481 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
62, 5mpbird 258 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → dom 𝐹 ∈ V)
7 fnfun 6446 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
87adantr 481 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → Fun 𝐹)
9 funrnex 7644 . . . . 5 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
106, 8, 9sylc 65 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ∈ V)
112, 10xpexd 7463 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12 simpl 483 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 Fn 𝐴)
13 dffn3 6518 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1412, 13sylib 219 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
15 fssxp 6527 . . . 4 (𝐹:𝐴⟶ran 𝐹𝐹 ⊆ (𝐴 × ran 𝐹))
1614, 15syl 17 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
17 ssdomg 8543 . . 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 8602 . . . . 5 ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
2010, 19sylancom 588 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
21 omex 9094 . . . . 5 ω ∈ V
22 fnrndomg 9946 . . . . . . 7 (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
232, 12, 22sylc 65 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹𝐴)
24 domtr 8550 . . . . . 6 ((ran 𝐹𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
2523, 24sylancom 588 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26 xpdom2g 8601 . . . . 5 ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
2721, 25, 26sylancr 587 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
28 domtr 8550 . . . 4 (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
2920, 27, 28syl2anc 584 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
30 xpomen 9429 . . 3 (ω × ω) ≈ ω
31 domentr 8556 . . 3 (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
3229, 30, 31sylancl 586 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
33 domtr 8550 . 2 ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
3418, 32, 33syl2anc 584 1 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492  wss 3933   class class class wbr 5057   × cxp 5546  dom cdm 5548  ran crn 5549  Fun wfun 6342   Fn wfn 6343  wf 6344  ωcom 7569  cen 8494  cdom 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356  df-acn 9359  df-ac 9530
This theorem is referenced by:  mptct  9948  mpocti  30377  mptctf  30379  omssubadd  31457
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