unirnffid - Metamath Proof Explorer

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Theorem unirnffid 9226

Description: The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)

**Hypotheses**
Ref	Expression
unirnffid.1	⊢ (𝜑 → 𝐹:𝑇⟶Fin)
unirnffid.2	⊢ (𝜑 → 𝑇 ∈ Fin)

**Assertion**
Ref	Expression
unirnffid	⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)

**Proof of Theorem unirnffid**
Step	Hyp	Ref	Expression
1		unirnffid.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝑇⟶Fin)
2	1	ffnd 6648	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnffid.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		fnfi 9082	. . . 4 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ Fin) → 𝐹 ∈ Fin)
5	2, 3, 4	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐹 ∈ Fin)
6		rnfi 9219	. . 3 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
8	1	frnd 6655	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ Fin)
9		unifi 9223	. 2 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ∪ ran 𝐹 ∈ Fin)
10	7, 8, 9	syl2anc 584	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3900 ∪ cuni 4857 ran crn 5615 Fn wfn 6472 ⟶wf 6473 Fincfn 8864

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-om 7792 df-1st 7916 df-2nd 7917 df-1o 8380 df-en 8865 df-dom 8866 df-fin 8868

This theorem is referenced by: marypha2 9318 acsinfd 18454

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