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Theorem fnct 28876
 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 7856 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
21adantl 481 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐴 ∈ V)
3 fndm 5904 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43eleq1d 2672 . . . . . . 7 (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
54adantr 480 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
62, 5mpbird 246 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → dom 𝐹 ∈ V)
7 fnfun 5902 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
87adantr 480 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → Fun 𝐹)
9 funrnex 7026 . . . . 5 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
106, 8, 9sylc 63 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ∈ V)
11 xpexg 6858 . . . 4 ((𝐴 ∈ V ∧ ran 𝐹 ∈ V) → (𝐴 × ran 𝐹) ∈ V)
122, 10, 11syl2anc 691 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
13 simpl 472 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 Fn 𝐴)
14 dffn3 5967 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1513, 14sylib 207 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
16 fssxp 5973 . . . 4 (𝐹:𝐴⟶ran 𝐹𝐹 ⊆ (𝐴 × ran 𝐹))
1715, 16syl 17 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
18 ssdomg 7887 . . 3 ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
1912, 17, 18sylc 63 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
20 xpdom1g 7942 . . . . 5 ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
2110, 20sylancom 698 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
22 omex 8423 . . . . 5 ω ∈ V
23 fnrndomg 9239 . . . . . . 7 (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
242, 13, 23sylc 63 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹𝐴)
25 domtr 7895 . . . . . 6 ((ran 𝐹𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
2624, 25sylancom 698 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
27 xpdom2g 7941 . . . . 5 ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
2822, 26, 27sylancr 694 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
29 domtr 7895 . . . 4 (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
3021, 28, 29syl2anc 691 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
31 xpomen 8721 . . 3 (ω × ω) ≈ ω
32 domentr 7901 . . 3 (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
3330, 31, 32sylancl 693 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
34 domtr 7895 . 2 ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
3519, 33, 34syl2anc 691 1 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ωcom 6957   ≈ cen 7838   ≼ cdom 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822 This theorem is referenced by:  mptct  28880  mpt2cti  28881  mptctf  28883  omssubadd  29689
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