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Theorem fnct 10578
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 9005 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
21adantl 481 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐴 ∈ V)
3 fndm 6670 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43eleq1d 2825 . . . . . . 7 (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
54adantr 480 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
62, 5mpbird 257 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → dom 𝐹 ∈ V)
7 fnfun 6667 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
87adantr 480 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → Fun 𝐹)
9 funrnex 7979 . . . . 5 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
106, 8, 9sylc 65 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ∈ V)
112, 10xpexd 7772 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
12 simpl 482 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 Fn 𝐴)
13 dffn3 6747 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1412, 13sylib 218 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
15 fssxp 6762 . . . 4 (𝐹:𝐴⟶ran 𝐹𝐹 ⊆ (𝐴 × ran 𝐹))
1614, 15syl 17 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
17 ssdomg 9041 . . 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 9110 . . . . 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 9684 . . . . 5 ω ∈ V
22 fnrndomg 10577 . . . . . . 7 (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
232, 12, 22sylc 65 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹𝐴)
24 domtr 9048 . . . . . 6 ((ran 𝐹𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
2523, 24sylancom 588 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
26 xpdom2g 9109 . . . . 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 9048 . . . 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 10056 . . 3 (ω × ω) ≈ ω
31 domentr 9054 . . 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 9048 . 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 206  wa 395  wcel 2107  Vcvv 3479  wss 3950   class class class wbr 5142   × cxp 5682  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  wf 6556  ωcom 7888  cen 8983  cdom 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-oi 9551  df-card 9980  df-acn 9983  df-ac 10157
This theorem is referenced by:  mptct  10579  mpocti  32728  mptctf  32730  omssubadd  34303
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