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Theorem unirnfdomd 10607
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)
21ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn 𝑇)
3 unirnfdomd.3 . . . . . . 7 (𝜑𝑇𝑉)
4 fnex 7237 . . . . . . 7 ((𝐹 Fn 𝑇𝑇𝑉) → 𝐹 ∈ V)
52, 3, 4syl2anc 584 . . . . . 6 (𝜑𝐹 ∈ V)
6 rnexg 7924 . . . . . 6 (𝐹 ∈ V → ran 𝐹 ∈ V)
75, 6syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ V)
8 frn 6743 . . . . . . 7 (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
9 dfss3 3972 . . . . . . 7 (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
108, 9sylib 218 . . . . . 6 (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11 fict 9693 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ≼ ω)
1211ralimi 3083 . . . . . 6 (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
131, 10, 123syl 18 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
14 unidom 10583 . . . . 5 ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ran 𝐹 ≼ (ran 𝐹 × ω))
157, 13, 14syl2anc 584 . . . 4 (𝜑 ran 𝐹 ≼ (ran 𝐹 × ω))
16 fnrndomg 10576 . . . . . 6 (𝑇𝑉 → (𝐹 Fn 𝑇 → ran 𝐹𝑇))
173, 2, 16sylc 65 . . . . 5 (𝜑 → ran 𝐹𝑇)
18 omex 9683 . . . . . 6 ω ∈ V
1918xpdom1 9111 . . . . 5 (ran 𝐹𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
2017, 19syl 17 . . . 4 (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
21 domtr 9047 . . . 4 (( ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ran 𝐹 ≼ (𝑇 × ω))
2215, 20, 21syl2anc 584 . . 3 (𝜑 ran 𝐹 ≼ (𝑇 × ω))
23 unirnfdomd.2 . . . . 5 (𝜑 → ¬ 𝑇 ∈ Fin)
24 infinf 10606 . . . . . 6 (𝑇𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
253, 24syl 17 . . . . 5 (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
2623, 25mpbid 232 . . . 4 (𝜑 → ω ≼ 𝑇)
27 xpdom2g 9108 . . . 4 ((𝑇𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
283, 26, 27syl2anc 584 . . 3 (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
29 domtr 9047 . . 3 (( ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ran 𝐹 ≼ (𝑇 × 𝑇))
3022, 28, 29syl2anc 584 . 2 (𝜑 ran 𝐹 ≼ (𝑇 × 𝑇))
31 infxpidm 10602 . . 3 (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
3226, 31syl 17 . 2 (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
33 domentr 9053 . 2 (( ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ran 𝐹𝑇)
3430, 32, 33syl2anc 584 1 (𝜑 ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wcel 2108  wral 3061  Vcvv 3480  wss 3951   cuni 4907   class class class wbr 5143   × cxp 5683  ran crn 5686   Fn wfn 6556  wf 6557  ωcom 7887  cen 8982  cdom 8983  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979  df-acn 9982  df-ac 10156
This theorem is referenced by:  acsdomd  18602
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