Theorem edgvalOLD 25942

Description: Obsolete version of edgval 25941 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 25940	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2	1	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3		fveq2 6191	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
4	3	rneqd 5353	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
5	4	adantl 482	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6		elex 3212	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		fvex 6201	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
8	7	rnex 7100	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑉 → ran (iEdg‘𝐺) ∈ V)
10	2, 5, 6, 9	fvmptd 6288	1 ⊢ (𝐺 ∈ 𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ↦ cmpt 4729  ran crn 5115  ‘cfv 5888  iEdgciedg 25875  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-edg 25940

This theorem is referenced by:  edgiedgbOLD  25948  edg0iedg0OLD  25950  edginwlkOLD  26531