Theorem unirnfdomd 10676

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 6729	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnfdomd.3	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
4		fnex 7243	. . . . . . 7 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ 𝑉) → 𝐹 ∈ V)
5	2, 3, 4	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
6		rnexg 7935	. . . . . 6 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V)
7	5, 6	syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ∈ V)
8		frn 6736	. . . . . . 7 ⊢ (𝐹:𝑇⟶Fin → ran 𝐹 ⊆ Fin)
9		dfss3 3902	. . . . . . 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 9678	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ≼ ω)
12	11	ralimi 3103	. . . . . 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 10651	. . . . 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 10642	. . . . . 6 ⊢ (𝑇 ∈ 𝑉 → (𝐹 Fn 𝑇 → ran 𝐹 ≼ 𝑇))
17	3, 2, 16	sylc 66	. . . . 5 ⊢ (𝜑 → ran 𝐹 ≼ 𝑇)
18		omex 9668	. . . . . 6 ⊢ ω ∈ V
19	18	xpdom1 9120	. . . . 5 ⊢ (ran 𝐹 ≼ 𝑇 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
20	17, 19	syl 18	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ω) ≼ (𝑇 × ω))
21		domtr 9059	. . . 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 10675	. . . . . 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 9117	. . . 4 ⊢ ((𝑇 ∈ 𝑉 ∧ ω ≼ 𝑇) → (𝑇 × ω) ≼ (𝑇 × 𝑇))
28	3, 26, 27	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑇 × ω) ≼ (𝑇 × 𝑇))
29		domtr 9059	. . 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 10670	. . 3 ⊢ (ω ≼ 𝑇 → (𝑇 × 𝑇) ≈ 𝑇)
32	26, 31	syl 18	. 2 ⊢ (𝜑 → (𝑇 × 𝑇) ≈ 𝑇)
33		domentr 9065	. 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 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  ∪ cuni 4870   class class class wbr 5106   × cxp 5653  ran crn 5656   Fn wfn 6550  ⟶wf 6551  ωcom 7898   ≈ cen 8995   ≼ cdom 8996  Fincfn 8998

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-ac2 10565

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-oi 9528  df-card 10044  df-acn 10047  df-ac 10219

This theorem is used by:  acsdomd  18757