unirnffid - Metamath Proof Explorer

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Theorem unirnffid 9280

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 6681	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝑇)
3		unirnffid.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		fnfi 9135	. . . 4 ⊢ ((𝐹 Fn 𝑇 ∧ 𝑇 ∈ Fin) → 𝐹 ∈ Fin)
5	2, 3, 4	syl2anc 592	. . 3 ⊢ (𝜑 → 𝐹 ∈ Fin)
6		rnfi 9273	. . 3 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
8	1	frnd 6689	. 2 ⊢ (𝜑 → ran 𝐹 ⊆ Fin)
9		unifi 9277	. 2 ⊢ ((ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin) → ∪ ran 𝐹 ∈ Fin)
10	7, 8, 9	syl2anc 592	1 ⊢ (𝜑 → ∪ ran 𝐹 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2136 ⊆ wss 3899 ∪ cuni 4859 ran crn 5641 Fn wfn 6505 ⟶wf 6506 Fincfn 8916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-om 7836 df-1st 7959 df-2nd 7960 df-1o 8425 df-en 8917 df-dom 8918 df-fin 8920

This theorem is referenced by: marypha2 9375 acsinfd 18564

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