Theorem unirnfdomd 9389

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 6045	. . . . . . . 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 6481	. . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V)
6	3, 4, 5	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
7		rnexg 7098	. . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V)
9		frn 6053	. . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
10		dfss3 3592	. . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11	9, 10	sylib 208	. . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
12		isfinite 8549	. . . . . . . 8 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
13		sdomdom 7983	. . . . . . . 8 ⊢ (𝑥 ≺ ω → 𝑥 ≼ ω)
14	12, 13	sylbi 207	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω)
15	14	ralimi 2952	. . . . . 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 9365	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
18	8, 16, 17	syl2anc 693	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
19		fnrndomg 9358	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇))
20	4, 3, 19	sylc 65	. . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇)
21		omex 8540	. . . . . 6 ⊢ ω ∈ V
22	21	xpdom1 8059	. . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
23	20, 22	syl 17	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
24		domtr 8009	. . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω))
25	18, 23, 24	syl2anc 693	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω))
26		unirnfdomd.2	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
27		infinf 9388	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
28	4, 27	syl 17	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
29	26, 28	mpbid 222	. . . 4 ⊢ (𝜑 → ω ≼ 𝑇)
30		xpdom2g 8056	. . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
31	4, 29, 30	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
32		domtr 8009	. . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
33	25, 31, 32	syl2anc 693	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
34		infxpidm 9384	. . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
35	29, 34	syl 17	. 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
36		domentr 8015	. 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇)
37	33, 35, 36	syl2anc 693	1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∪ cuni 4436   class class class wbr 4653   × cxp 5112  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ωcom 7065   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by:  acsdomd  17181