Theorem edgopval 25944

Description: The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)

Assertion

Ref	Expression
edgopval	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Proof of Theorem edgopval

Step	Hyp	Ref	Expression
1		edgval 25941	. 2 ⊢ (Edg‘⟨𝑉, 𝐸⟩) = ran (iEdg‘⟨𝑉, 𝐸⟩)
2		opiedgfv 25887	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
3	2	rneqd 5353	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
4	1, 3	syl5eq 2668	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ⟨cop 4183  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-pow 4843  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-2nd 7169  df-iedg 25877  df-edg 25940

This theorem is referenced by:  edgov  25945  cusgrsize  26350  uspgrloopedg  26414  uspgrsprfo  41756