Theorem edgvalOLD 26162

Description: Obsolete version of edgval 26161 as of 8-Dec-2021. (Contributed by AV, 1-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
edgvalOLD	⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Proof of Theorem edgvalOLD

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-edg 26160	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2	1	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3		fveq2 6353	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
4	3	rneqd 5508	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
5	4	adantl 473	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6		elex 3352	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		fvex 6363	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
8	7	rnex 7266	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑉 → ran (iEdg‘𝐺) ∈ V)
10	2, 5, 6, 9	fvmptd 6451	1 ⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ↦ cmpt 4881  ran crn 5267  ‘cfv 6049  iEdgciedg 26095  Edgcedg 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-edg 26160

This theorem is referenced by:  edgiedgbOLD  26168  edg0iedg0OLD  26170  edginwlkOLD  26762