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Theorem unirnfdomd 9334
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 6004 . . . . . . . 8 (𝐹:𝑇⟶Fin → 𝐹 Fn 𝑇)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝑇)
4 unirnfdomd.3 . . . . . . 7 (𝜑𝑇𝑉)
5 fnex 6436 . . . . . . 7 ((𝐹 Fn 𝑇𝑇𝑉) → 𝐹 ∈ V)
63, 4, 5syl2anc 692 . . . . . 6 (𝜑𝐹 ∈ V)
7 rnexg 7046 . . . . . 6 (𝐹 ∈ V → ran 𝐹 ∈ V)
86, 7syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ V)
9 frn 6012 . . . . . . 7 (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
10 dfss3 3578 . . . . . . 7 (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
119, 10sylib 208 . . . . . 6 (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
12 isfinite 8494 . . . . . . . 8 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
13 sdomdom 7928 . . . . . . . 8 (𝑥 ≺ ω → 𝑥 ≼ ω)
1412, 13sylbi 207 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ≼ ω)
1514ralimi 2952 . . . . . 6 (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
161, 11, 153syl 18 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
17 unidom 9310 . . . . 5 ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ran 𝐹 ≼ (ran 𝐹 × ω))
188, 16, 17syl2anc 692 . . . 4 (𝜑 ran 𝐹 ≼ (ran 𝐹 × ω))
19 fnrndomg 9303 . . . . . 6 (𝑇𝑉 → (𝐹 Fn 𝑇 → ran 𝐹𝑇))
204, 3, 19sylc 65 . . . . 5 (𝜑 → ran 𝐹𝑇)
21 omex 8485 . . . . . 6 ω ∈ V
2221xpdom1 8004 . . . . 5 (ran 𝐹𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
2320, 22syl 17 . . . 4 (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
24 domtr 7954 . . . 4 (( ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ran 𝐹 ≼ (𝑇 × ω))
2518, 23, 24syl2anc 692 . . 3 (𝜑 ran 𝐹 ≼ (𝑇 × ω))
26 unirnfdomd.2 . . . . 5 (𝜑 → ¬ 𝑇 ∈ Fin)
27 infinf 9333 . . . . . 6 (𝑇𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
284, 27syl 17 . . . . 5 (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
2926, 28mpbid 222 . . . 4 (𝜑 → ω ≼ 𝑇)
30 xpdom2g 8001 . . . 4 ((𝑇𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
314, 29, 30syl2anc 692 . . 3 (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
32 domtr 7954 . . 3 (( ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ran 𝐹 ≼ (𝑇 × 𝑇))
3325, 31, 32syl2anc 692 . 2 (𝜑 ran 𝐹 ≼ (𝑇 × 𝑇))
34 infxpidm 9329 . . 3 (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
3529, 34syl 17 . 2 (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
36 domentr 7960 . 2 (( ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ran 𝐹𝑇)
3733, 35, 36syl2anc 692 1 (𝜑 ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 1992  wral 2912  Vcvv 3191  wss 3560   cuni 4407   class class class wbr 4618   × cxp 5077  ran crn 5080   Fn wfn 5845  wf 5846  ωcom 7013  cen 7897  cdom 7898  csdm 7899  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-ac2 9230
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-card 8710  df-acn 8713  df-ac 8884
This theorem is referenced by:  acsdomd  17097
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