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Theorem vtxdgfisf 27256

Description: The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)

**Hypotheses**
Ref	Expression
vtxdgf.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdg0e.i	⊢ 𝐼 = (iEdg‘𝐺)
vtxdgfisnn0.a	⊢ 𝐴 = dom 𝐼

**Assertion**
Ref	Expression
vtxdgfisf	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀)

Proof of Theorem vtxdgfisf Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgf.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxdgf 27251	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
3	2	adantr 483	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀^*)
4	3	ffnd 6508	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺) Fn 𝑉)
5		vtxdg0e.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
6		vtxdgfisnn0.a	. . . . 5 ⊢ 𝐴 = dom 𝐼
7	1, 5, 6	vtxdgfisnn0 27255	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
8	7	adantll 712	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) ∧ 𝑢 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
9	8	ralrimiva 3181	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀)
10		ffnfv 6875	. 2 ⊢ ((VtxDeg‘𝐺):𝑉⟶ℕ₀ ↔ ((VtxDeg‘𝐺) Fn 𝑉 ∧ ∀𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) ∈ ℕ₀))
11	4, 9, 10	sylanbrc 585	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 dom cdm 5548 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 Fincfn 8502 ℕ₀cn0 11891 ℕ₀^*cxnn0 11961 Vtxcvtx 26779 iEdgciedg 26780 VtxDegcvtxdg 27245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 df-hash 13688 df-vtxdg 27246

This theorem is referenced by: vtxdgfusgrf 27277 eupth2lem3lem1 28005 eupth2lem3lem2 28006

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