unirnffid - Metamath Proof Explorer

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Theorem unirnffid 9250

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 6659	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnffid.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		fnfi 9105	. . . 4 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ Fin) → 𝐹 ∈ Fin)
5	2, 3, 4	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ Fin)
6		rnfi 9243	. . 3 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
8	1	frnd 6666	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ Fin)
9		unifi 9247	. 2 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ∪ ran 𝐹 ∈ Fin)
10	7, 8, 9	syl2anc 586	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2115 ⊆ wss 3886 ∪ cuni 4841 ran crn 5622 Fn wfn 6483 ⟶wf 6484 Fincfn 8886

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1970 ax-7 2011 ax-8 2117 ax-9 2125 ax-10 2148 ax-11 2164 ax-12 2185 ax-ext 2708 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7681

This theorem depends on definitions: df-bi 208 df-an 397 df-or 850 df-3or 1089 df-3an 1090 df-tru 1546 df-fal 1556 df-ex 1783 df-nf 1787 df-sb 2070 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2932 df-ral 3051 df-rex 3061 df-reu 3342 df-rab 3389 df-v 3430 df-sbc 3727 df-csb 3835 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3906 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-om 7810 df-1st 7934 df-2nd 7935 df-1o 8398 df-en 8887 df-dom 8888 df-fin 8890

This theorem is referenced by: marypha2 9345 acsinfd 18516

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