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Theorem unirnfdomd 9427
Description: The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
unirnfdomd.1 (𝜑𝐹:𝑇⟶Fin)
unirnfdomd.2 (𝜑 → ¬ 𝑇 ∈ Fin)
unirnfdomd.3 (𝜑𝑇𝑉)
Assertion
Ref Expression
unirnfdomd (𝜑 ran 𝐹𝑇)

Proof of Theorem unirnfdomd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unirnfdomd.1 . . . . . . . 8 (𝜑𝐹:𝑇⟶Fin)
2 ffn 6083 . . . . . . . 8 (𝐹:𝑇⟶Fin → 𝐹 Fn 𝑇)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝑇)
4 unirnfdomd.3 . . . . . . 7 (𝜑𝑇𝑉)
5 fnex 6522 . . . . . . 7 ((𝐹 Fn 𝑇𝑇𝑉) → 𝐹 ∈ V)
63, 4, 5syl2anc 694 . . . . . 6 (𝜑𝐹 ∈ V)
7 rnexg 7140 . . . . . 6 (𝐹 ∈ V → ran 𝐹 ∈ V)
86, 7syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ V)
9 frn 6091 . . . . . . 7 (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
10 dfss3 3625 . . . . . . 7 (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
119, 10sylib 208 . . . . . 6 (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
12 isfinite 8587 . . . . . . . 8 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
13 sdomdom 8025 . . . . . . . 8 (𝑥 ≺ ω → 𝑥 ≼ ω)
1412, 13sylbi 207 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ≼ ω)
1514ralimi 2981 . . . . . 6 (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
161, 11, 153syl 18 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
17 unidom 9403 . . . . 5 ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ran 𝐹 ≼ (ran 𝐹 × ω))
188, 16, 17syl2anc 694 . . . 4 (𝜑 ran 𝐹 ≼ (ran 𝐹 × ω))
19 fnrndomg 9396 . . . . . 6 (𝑇𝑉 → (𝐹 Fn 𝑇 → ran 𝐹𝑇))
204, 3, 19sylc 65 . . . . 5 (𝜑 → ran 𝐹𝑇)
21 omex 8578 . . . . . 6 ω ∈ V
2221xpdom1 8100 . . . . 5 (ran 𝐹𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
2320, 22syl 17 . . . 4 (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
24 domtr 8050 . . . 4 (( ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ran 𝐹 ≼ (𝑇 × ω))
2518, 23, 24syl2anc 694 . . 3 (𝜑 ran 𝐹 ≼ (𝑇 × ω))
26 unirnfdomd.2 . . . . 5 (𝜑 → ¬ 𝑇 ∈ Fin)
27 infinf 9426 . . . . . 6 (𝑇𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
284, 27syl 17 . . . . 5 (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
2926, 28mpbid 222 . . . 4 (𝜑 → ω ≼ 𝑇)
30 xpdom2g 8097 . . . 4 ((𝑇𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
314, 29, 30syl2anc 694 . . 3 (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
32 domtr 8050 . . 3 (( ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ran 𝐹 ≼ (𝑇 × 𝑇))
3325, 31, 32syl2anc 694 . 2 (𝜑 ran 𝐹 ≼ (𝑇 × 𝑇))
34 infxpidm 9422 . . 3 (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
3529, 34syl 17 . 2 (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
36 domentr 8056 . 2 (( ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ran 𝐹𝑇)
3733, 35, 36syl2anc 694 1 (𝜑 ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 2030  wral 2941  Vcvv 3231  wss 3607   cuni 4468   class class class wbr 4685   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922  ωcom 7107  cen 7994  cdom 7995  csdm 7996  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977
This theorem is referenced by:  acsdomd  17228
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