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Theorem fnct 9543
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 8128 . . . . 5 (𝐴 ≼ ω → 𝐴 ∈ V)
21adantl 473 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐴 ∈ V)
3 fndm 6143 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43eleq1d 2816 . . . . . . 7 (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
54adantr 472 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
62, 5mpbird 247 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → dom 𝐹 ∈ V)
7 fnfun 6141 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
87adantr 472 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → Fun 𝐹)
9 funrnex 7290 . . . . 5 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
106, 8, 9sylc 65 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ∈ V)
11 xpexg 7117 . . . 4 ((𝐴 ∈ V ∧ ran 𝐹 ∈ V) → (𝐴 × ran 𝐹) ∈ V)
122, 10, 11syl2anc 696 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ∈ V)
13 simpl 474 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 Fn 𝐴)
14 dffn3 6207 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1513, 14sylib 208 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹:𝐴⟶ran 𝐹)
16 fssxp 6213 . . . 4 (𝐹:𝐴⟶ran 𝐹𝐹 ⊆ (𝐴 × ran 𝐹))
1715, 16syl 17 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ⊆ (𝐴 × ran 𝐹))
18 ssdomg 8159 . . 3 ((𝐴 × ran 𝐹) ∈ V → (𝐹 ⊆ (𝐴 × ran 𝐹) → 𝐹 ≼ (𝐴 × ran 𝐹)))
1912, 17, 18sylc 65 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ (𝐴 × ran 𝐹))
20 xpdom1g 8214 . . . . 5 ((ran 𝐹 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
2110, 20sylancom 704 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ran 𝐹))
22 omex 8705 . . . . 5 ω ∈ V
23 fnrndomg 9542 . . . . . . 7 (𝐴 ∈ V → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
242, 13, 23sylc 65 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹𝐴)
25 domtr 8166 . . . . . 6 ((ran 𝐹𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
2624, 25sylancom 704 . . . . 5 ((𝐹 Fn 𝐴𝐴 ≼ ω) → ran 𝐹 ≼ ω)
27 xpdom2g 8213 . . . . 5 ((ω ∈ V ∧ ran 𝐹 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
2822, 26, 27sylancr 698 . . . 4 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (ω × ran 𝐹) ≼ (ω × ω))
29 domtr 8166 . . . 4 (((𝐴 × ran 𝐹) ≼ (ω × ran 𝐹) ∧ (ω × ran 𝐹) ≼ (ω × ω)) → (𝐴 × ran 𝐹) ≼ (ω × ω))
3021, 28, 29syl2anc 696 . . 3 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ (ω × ω))
31 xpomen 9020 . . 3 (ω × ω) ≈ ω
32 domentr 8172 . . 3 (((𝐴 × ran 𝐹) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ran 𝐹) ≼ ω)
3330, 31, 32sylancl 697 . 2 ((𝐹 Fn 𝐴𝐴 ≼ ω) → (𝐴 × ran 𝐹) ≼ ω)
34 domtr 8166 . 2 ((𝐹 ≼ (𝐴 × ran 𝐹) ∧ (𝐴 × ran 𝐹) ≼ ω) → 𝐹 ≼ ω)
3519, 33, 34syl2anc 696 1 ((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2131  Vcvv 3332  wss 3707   class class class wbr 4796   × cxp 5256  dom cdm 5258  ran crn 5259  Fun wfun 6035   Fn wfn 6036  wf 6037  ωcom 7222  cen 8110  cdom 8111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-ac2 9469
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-card 8947  df-acn 8950  df-ac 9121
This theorem is referenced by:  mptct  9544  mpt2cti  29794  mptctf  29796  omssubadd  30663
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