Theorem unirnfdomd 10486

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	⊢ (𝜑 → 𝐹:𝑇⟶Fin)
unirnfdomd.2	⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
unirnfdomd.3	⊢ (𝜑 → 𝑇 ∈ 𝑉)

Assertion

Ref	Expression
unirnfdomd	⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)

Proof of Theorem unirnfdomd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unirnfdomd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶Fin)
2	1	ffnd 6659	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnfdomd.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
4		fnex 7164	. . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V)
5	2, 3, 4	syl2anc 591	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
6		rnexg 7846	. . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V)
8		frn 6665	. . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
9		dfss3 3905	. . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
10	8, 9	sylib 220	. . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11		fict 9569	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω)
12	11	ralimi 3078	. . . . . 6 ⊢ (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
13	1, 10, 12	3syl 18	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
14		unidom 10461	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
15	7, 13, 14	syl2anc 591	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
16		fnrndomg 10454	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇))
17	3, 2, 16	sylc 65	. . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇)
18		omex 9559	. . . . . 6 ⊢ ω ∈ V
19	18	xpdom1 9008	. . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
20	17, 19	syl 17	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
21		domtr 8948	. . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω))
22	15, 20, 21	syl2anc 591	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω))
23		unirnfdomd.2	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
24		infinf 10485	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
25	3, 24	syl 17	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
26	23, 25	mpbid 234	. . . 4 ⊢ (𝜑 → ω ≼ 𝑇)
27		xpdom2g 9005	. . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
28	3, 26, 27	syl2anc 591	. . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
29		domtr 8948	. . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
30	22, 28, 29	syl2anc 591	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
31		infxpidm 10480	. . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
32	26, 31	syl 17	. 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
33		domentr 8954	. 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇)
34	30, 32, 33	syl2anc 591	1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3884  ∪ cuni 4840   class class class wbr 5074   × cxp 5618  ran crn 5621   Fn wfn 6483  ⟶wf 6484  ωcom 7809   ≈ cen 8884   ≼ cdom 8885  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-inf2 9557  ax-ac2 10381

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-card 9858  df-acn 9861  df-ac 10033

This theorem is referenced by:  acsdomd  18518