Theorem edgiedgb 29141

Description: A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)

Hypothesis

Ref	Expression
edgiedgb.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
edgiedgb	⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem edgiedgb

Step	Hyp	Ref	Expression
1		edgval 29136	. . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		edgiedgb.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	eqcomi 2746	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
4	3	rneqi 5888	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
5	1, 4	eqtri 2760	. . 3 ⊢ (Edg‘𝐺) = ran 𝐼
6	5	eleq2i 2829	. 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7		elrnrexdmb 7038	. 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
8	6, 7	bitrid 283	1 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  dom cdm 5626  ran crn 5627  Fun wfun 6488  ‘cfv 6494  iEdgciedg 29084  Edgcedg 29134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-iota 6450  df-fun 6496  df-fn 6497  df-fv 6502  df-edg 29135

This theorem is referenced by:  uhgredgiedgb  29213  isubgredg  48358  uhgrimedgi  48382