Theorem unirnfdomd 9268

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		ffn 5958	. . . . . . . 8 ⊢ (𝐹:𝑇⟶Fin → 𝐹 Fn 𝑇)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇)
4		unirnfdomd.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
5		fnex 6386	. . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V)
6	3, 4, 5	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
7		rnexg 6990	. . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V)
9		frn 5966	. . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
10		dfss3 3558	. . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11	9, 10	sylib 207	. . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
12		isfinite 8432	. . . . . . . 8 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
13		sdomdom 7869	. . . . . . . 8 ⊢ (𝑥 ≺ ω → 𝑥 ≼ ω)
14	12, 13	sylbi 206	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω)
15	14	ralimi 2936	. . . . . 6 ⊢ (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
16	1, 11, 15	3syl 18	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
17		unidom 9244	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
18	8, 16, 17	syl2anc 691	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
19		fnrndomg 9239	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇))
20	4, 3, 19	sylc 63	. . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇)
21		omex 8423	. . . . . 6 ⊢ ω ∈ V
22	21	xpdom1 7944	. . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
23	20, 22	syl 17	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
24		domtr 7895	. . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω))
25	18, 23, 24	syl2anc 691	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω))
26		unirnfdomd.2	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
27		infinf 9267	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
28	4, 27	syl 17	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
29	26, 28	mpbid 221	. . . 4 ⊢ (𝜑 → ω ≼ 𝑇)
30		xpdom2g 7941	. . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
31	4, 29, 30	syl2anc 691	. . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
32		domtr 7895	. . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
33	25, 31, 32	syl2anc 691	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
34		infxpidm 9263	. . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
35	29, 34	syl 17	. 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
36		domentr 7901	. 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇)
37	33, 35, 36	syl2anc 691	1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822

This theorem is referenced by:  acsdomd  17004