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Theorem unirnfdomd 10605
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 6738 . . . . . . 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 7925 . . . . . 6 (𝐹 ∈ V → ran 𝐹 ∈ V)
75, 6syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ V)
8 frn 6744 . . . . . . 7 (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
9 dfss3 3984 . . . . . . 7 (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
108, 9sylib 218 . . . . . 6 (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11 fict 9691 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ≼ ω)
1211ralimi 3081 . . . . . 6 (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
131, 10, 123syl 18 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
14 unidom 10581 . . . . 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 10574 . . . . . 6 (𝑇𝑉 → (𝐹 Fn 𝑇 → ran 𝐹𝑇))
173, 2, 16sylc 65 . . . . 5 (𝜑 → ran 𝐹𝑇)
18 omex 9681 . . . . . 6 ω ∈ V
1918xpdom1 9110 . . . . 5 (ran 𝐹𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
2017, 19syl 17 . . . 4 (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
21 domtr 9046 . . . 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 10604 . . . . . 6 (𝑇𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
253, 24syl 17 . . . . 5 (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
2623, 25mpbid 232 . . . 4 (𝜑 → ω ≼ 𝑇)
27 xpdom2g 9107 . . . 4 ((𝑇𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
283, 26, 27syl2anc 584 . . 3 (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
29 domtr 9046 . . 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 10600 . . 3 (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
3226, 31syl 17 . 2 (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
33 domentr 9052 . 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 2106  wral 3059  Vcvv 3478  wss 3963   cuni 4912   class class class wbr 5148   × cxp 5687  ran crn 5690   Fn wfn 6558  wf 6559  ωcom 7887  cen 8981  cdom 8982  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977  df-acn 9980  df-ac 10154
This theorem is referenced by:  acsdomd  18615
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