Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isubgredg Structured version   Visualization version   GIF version

Theorem isubgredg 47852
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 2737 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
53, 4isubgriedg 47849 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
62, 5eqtrid 2789 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
76rneqd 5949 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆}))
87eleq2d 2827 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆𝑉) → (𝐾 ∈ ran (iEdg‘𝐻) ↔ 𝐾 ∈ ran ((iEdg‘𝐺) ↾ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆})))
93, 4uhgrf 29079 . . . . . . 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 4080 . . . . . 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 6969 . . . 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 2739 . . . . . 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 4015 . . . . . . . . . 10 (𝑖 = 𝑥 → (((iEdg‘𝐺)‘𝑖) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
2221elrab 3692 . . . . . . . . 9 (𝑥 ∈ {𝑖 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑆} ↔ (𝑥 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆))
234uhgrfun 29083 . . . . . . . . . . . . 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 7096 . . . . . . . . . . . 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 2829 . . . . . . . 8 (((iEdg‘𝐺)‘𝑥) = 𝐾 → (((iEdg‘𝐺)‘𝑥) ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ ran (iEdg‘𝐺)))
35 sseq1 4009 . . . . . . . 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 3155 . . . 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 29066 . . . . . . . . . . 11 (Edg‘𝐺) = ran (iEdg‘𝐺)
4140eqcomi 2746 . . . . . . . . . 10 ran (iEdg‘𝐺) = (Edg‘𝐺)
4241eleq2i 2833 . . . . . . . . 9 (𝐾 ∈ ran (iEdg‘𝐺) ↔ 𝐾 ∈ (Edg‘𝐺))
434edgiedgb 29071 . . . . . . . . 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 4015 . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . 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 1917 . . . . . . . . 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 3071 . . . . . . . . 9 (∃𝑥 ∈ dom (iEdg‘𝐺)𝐾 = ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑥(𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐾 = ((iEdg‘𝐺)‘𝑥)))
63 df-rex 3071 . . . . . . . . 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 29066 . . . 4 (Edg‘𝐻) = ran (iEdg‘𝐻)
7371, 72eqtri 2765 . . 3 𝐼 = ran (iEdg‘𝐻)
7473eleq2i 2833 . 2 (𝐾𝐼𝐾 ∈ ran (iEdg‘𝐻))
75 isubgredg.e . . . . 5 𝐸 = (Edg‘𝐺)
7675, 40eqtri 2765 . . . 4 𝐸 = ran (iEdg‘𝐺)
7776eleq2i 2833 . . 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 1540  wex 1779  wcel 2108  wrex 3070  {crab 3436  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073   ISubGr cisubgr 47846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-isubgr 47847
This theorem is referenced by:  isubgr3stgrlem6  47938  isubgr3stgrlem7  47939  isubgr3stgrlem8  47940
  Copyright terms: Public domain W3C validator