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Theorem isubgredg 47789
Description: An edge of an induced subgraph of a hypergraph is an edge of the hypergraph connecting vertices of the subgraph. (Contributed by AV, 24-Sep-2025.)
Hypotheses
Ref Expression
isubgredg.v 𝑉 = (Vtx‘𝐺)
isubgredg.e 𝐸 = (Edg‘𝐺)
isubgredg.h 𝐻 = (𝐺 ISubGr 𝑆)
isubgredg.i 𝐼 = (Edg‘𝐻)
Assertion
Ref Expression
isubgredg ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾𝐼 ↔ (𝐾𝐸𝐾𝑆)))

Proof of Theorem isubgredg
Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isubgredg.h . . . . . . 7 𝐻 = (𝐺 ISubGr 𝑆)
21fveq2i 6909 . . . . . 6 (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3 isubgredg.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
4 eqid 2734 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
53, 4isubgriedg 47786 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
62, 5eqtrid 2786 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
76rneqd 5951 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
87eleq2d 2824 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ 𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})))
93, 4uhgrf 29093 . . . . . . 7 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
109adantr 480 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
1110ffnd 6737 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12 ssrab2 4089 . . . . . 6 {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)
1312a1i 11 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺))
1411, 13fnssresd 6692 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) Fn {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
15 fvelrnb 6968 . . . 4 (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) Fn {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → (𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) ↔ ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
1614, 15syl 17 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) ↔ ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
17 fvres 6925 . . . . . . . 8 (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = ((iEdg‘𝐺)‘𝑥))
1817adantl 481 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = ((iEdg‘𝐺)‘𝑥))
1918eqeq1d 2736 . . . . . 6 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → ((((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ ((iEdg‘𝐺)‘𝑥) = 𝐾))
20 fveq2 6906 . . . . . . . . . . 11 (𝑖 = 𝑥 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑥))
2120sseq1d 4026 . . . . . . . . . 10 (𝑖 = 𝑥 → (((iEdg‘𝐺)‘𝑖) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
2221elrab 3694 . . . . . . . . 9 (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ↔ (𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
234uhgrfun 29097 . . . . . . . . . . . . 13 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
2423adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → Fun (iEdg‘𝐺))
25 simpl 482 . . . . . . . . . . . 12 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → 𝑥 ∈ dom (iEdg‘𝐺))
26 fvelrn 7095 . . . . . . . . . . . 12 ((Fun (iEdg‘𝐺) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
2724, 25, 26syl2anr 597 . . . . . . . . . . 11 (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆𝑉)) → ((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺))
28 simpr 484 . . . . . . . . . . . 12 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
2928adantr 480 . . . . . . . . . . 11 (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆𝑉)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
3027, 29jca 511 . . . . . . . . . 10 (((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ∧ (𝐺 ∈ UHGraph ∧ 𝑆𝑉)) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
3130ex 412 . . . . . . . . 9 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) → ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)))
3222, 31sylbi 217 . . . . . . . 8 (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} → ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)))
3332impcom 407 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
34 eleq1 2826 . . . . . . . 8 (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ ran (iEdg‘𝐺)))
35 sseq1 4020 . . . . . . . 8 (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆𝐾𝑆))
3634, 35anbi12d 632 . . . . . . 7 (((iEdg‘𝐺)‘𝑥) = 𝐾 → ((((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
3733, 36syl5ibcom 245 . . . . . 6 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺)‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
3819, 37sylbid 240 . . . . 5 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → ((((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
3938rexlimdva 3152 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 → (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
40 edgval 29080 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
4140eqcomi 2743 . . . . . . . . . 10 ran (iEdg‘𝐺) = (Edg‘𝐺)
4241eleq2i 2830 . . . . . . . . 9 (𝐾 ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ (Edg‘𝐺))
434edgiedgb 29085 . . . . . . . . 9 (Fun (iEdg‘𝐺) → (𝐾 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
4442, 43bitrid 283 . . . . . . . 8 (Fun (iEdg‘𝐺) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
4523, 44syl 17 . . . . . . 7 (𝐺 ∈ UHGraph → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
4645adantr 480 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran (iEdg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥)))
47 simprl 771 . . . . . . . . . . . . 13 ((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → 𝑥 ∈ dom (iEdg‘𝐺))
48 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → 𝐾 = ((iEdg‘𝐺)‘𝑥))
4948sseq1d 4026 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → (𝐾𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
5049biimpcd 249 . . . . . . . . . . . . . . 15 (𝐾𝑆 → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
5150adantl 481 . . . . . . . . . . . . . 14 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
5251imp 406 . . . . . . . . . . . . 13 ((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆)
5347, 52, 22sylanbrc 583 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
54 simpr 484 . . . . . . . . . . . . 13 (((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})
5548eqcomd 2740 . . . . . . . . . . . . . . 15 ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
5655adantl 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → ((iEdg‘𝐺)‘𝑥) = 𝐾)
5717, 56sylan9eqr 2796 . . . . . . . . . . . . 13 (((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)
5854, 57jca 511 . . . . . . . . . . . 12 (((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) ∧ 𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
5953, 58mpdan 687 . . . . . . . . . . 11 ((((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) ∧ (𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥))) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
6059ex 412 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) → ((𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
6160eximdv 1914 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) → (∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)) → ∃𝑥(𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
62 df-rex 3068 . . . . . . . . 9 (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)))
63 df-rex 3068 . . . . . . . . 9 (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ ∃𝑥(𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ∧ (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
6461, 62, 633imtr4g 296 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑆𝑉) ∧ 𝐾𝑆) → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
6564ex 412 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾𝑆 → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
6665com23 86 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) → (𝐾𝑆 → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
6746, 66sylbid 240 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran (iEdg‘𝐺) → (𝐾𝑆 → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾)))
6867impd 410 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → ((𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆) → ∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾))
6939, 68impbid 212 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (∃𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} (((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})‘𝑥) = 𝐾 ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
708, 16, 693bitrd 305 . 2 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆)))
71 isubgredg.i . . . 4 𝐼 = (Edg‘𝐻)
72 edgval 29080 . . . 4 (Edg‘𝐻) = ran (iEdg‘𝐻)
7371, 72eqtri 2762 . . 3 𝐼 = ran (iEdg‘𝐻)
7473eleq2i 2830 . 2 (𝐾𝐼𝐾 ∈ ran (iEdg‘𝐻))
75 isubgredg.e . . . . 5 𝐸 = (Edg‘𝐺)
7675, 40eqtri 2762 . . . 4 𝐸 = ran (iEdg‘𝐺)
7776eleq2i 2830 . . 3 (𝐾𝐸𝐾 ∈ ran (iEdg‘𝐺))
7877anbi1i 624 . 2 ((𝐾𝐸𝐾𝑆) ↔ (𝐾 ∈ ran (iEdg‘𝐺) ∧ 𝐾𝑆))
7970, 74, 783bitr4g 314 1 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾𝐼 ↔ (𝐾𝐸𝐾𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  {crab 3432  cdif 3959  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630  dom cdm 5688  ran crn 5689  cres 5690  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UHGraphcuhgr 29087   ISubGr cisubgr 47783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8013  df-iedg 29030  df-edg 29079  df-uhgr 29089  df-isubgr 47784
This theorem is referenced by:  isubgr3stgrlem6  47873  isubgr3stgrlem7  47874  isubgr3stgrlem8  47875
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