Theorem unirnfdomd 10579

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 6707	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnfdomd.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
4		fnex 7219	. . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V)
5	2, 3, 4	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
6		rnexg 7902	. . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
7	5, 6	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V)
8		frn 6714	. . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
9		dfss3 3923	. . . . . . 7 ⊢ (ran 𝐹 ⊆ Fin ↔ ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
10	8, 9	sylib 221	. . . . . 6 ⊢ (𝐹:𝑇⟶Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin)
11		fict 9635	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω)
12	11	ralimi 3101	. . . . . 6 ⊢ (∀𝑥 ∈ ran 𝐹 𝑥 ∈ Fin → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
13	1, 10, 12	3syl 19	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω)
14		unidom 10554	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ ∀𝑥 ∈ ran 𝐹 𝑥 ≼ ω) → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
15	7, 13, 14	syl2anc 596	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (ran 𝐹 × ω))
16		fnrndomg 10545	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇))
17	3, 2, 16	sylc 66	. . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇)
18		omex 9625	. . . . . 6 ⊢ ω ∈ V
19	18	xpdom1 9077	. . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
20	17, 19	syl 18	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
21		domtr 9016	. . . 4 ⊢ ((∪ ran 𝐹 ≼ (ran 𝐹 × ω) ∧ (ran 𝐹 × ω) ≼ (𝑇 × ω)) → ∪ ran 𝐹 ≼ (𝑇 × ω))
22	15, 20, 21	syl2anc 596	. . 3 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × ω))
23		unirnfdomd.2	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
24		infinf 10578	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
25	3, 24	syl 18	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ∈ Fin ↔ ω ≼ 𝑇))
26	23, 25	mpbid 235	. . . 4 ⊢ (𝜑 → ω ≼ 𝑇)
27		xpdom2g 9074	. . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
28	3, 26, 27	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
29		domtr 9016	. . 3 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × ω) ∧ (𝑇 × ω) ≼ (𝑇 × 𝑇)) → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
30	22, 28, 29	syl2anc 596	. 2 ⊢ (𝜑 → ∪ ran 𝐹 ≼ (𝑇 × 𝑇))
31		infxpidm 10573	. . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
32	26, 31	syl 18	. 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
33		domentr 9022	. 2 ⊢ ((∪ ran 𝐹 ≼ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ≈ 𝑇) → ∪ ran 𝐹 ≼ 𝑇)
34	30, 32, 33	syl2anc 596	1 ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4870   class class class wbr 5107   × cxp 5657  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ωcom 7865   ≈ cen 8952   ≼ cdom 8953  Fincfn 8955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-ac2 10468

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-card 9947  df-acn 9950  df-ac 10122

This theorem is used by:  acsdomd  18649