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Theorem isomgreqve 47070
Description: A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.)
Assertion
Ref Expression
isomgreqve (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → š“ IsomGr šµ)

Proof of Theorem isomgreqve
Dummy variables š‘“ š‘” š‘– are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6900 . . . 4 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (Vtxā€˜šµ) ∈ V)
21resiexd 7213 . . 3 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → ( I ↾ (Vtxā€˜šµ)) ∈ V)
3 f1oi 6865 . . . . 5 ( I ↾ (Vtxā€˜šµ)):(Vtxā€˜šµ)–1-1-onto→(Vtxā€˜šµ)
4 simprl 768 . . . . . 6 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (Vtxā€˜š“) = (Vtxā€˜šµ))
54f1oeq2d 6823 . . . . 5 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (( I ↾ (Vtxā€˜šµ)):(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ↔ ( I ↾ (Vtxā€˜šµ)):(Vtxā€˜šµ)–1-1-onto→(Vtxā€˜šµ)))
63, 5mpbiri 258 . . . 4 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → ( I ↾ (Vtxā€˜šµ)):(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ))
7 fvexd 6900 . . . . . . 7 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (iEdgā€˜šµ) ∈ V)
87dmexd 7893 . . . . . 6 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → dom (iEdgā€˜šµ) ∈ V)
98resiexd 7213 . . . . 5 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → ( I ↾ dom (iEdgā€˜šµ)) ∈ V)
10 f1oi 6865 . . . . . . 7 ( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜šµ)–1-1-onto→dom (iEdgā€˜šµ)
11 simprr 770 . . . . . . . . 9 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (iEdgā€˜š“) = (iEdgā€˜šµ))
1211dmeqd 5899 . . . . . . . 8 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → dom (iEdgā€˜š“) = dom (iEdgā€˜šµ))
1312f1oeq2d 6823 . . . . . . 7 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ↔ ( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜šµ)–1-1-onto→dom (iEdgā€˜šµ)))
1410, 13mpbiri 258 . . . . . 6 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → ( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ))
15 eqid 2726 . . . . . . . . . . . 12 (Vtxā€˜š“) = (Vtxā€˜š“)
16 eqid 2726 . . . . . . . . . . . 12 (iEdgā€˜š“) = (iEdgā€˜š“)
1715, 16uhgrss 28832 . . . . . . . . . . 11 ((š“ ∈ UHGraph ∧ š‘– ∈ dom (iEdgā€˜š“)) → ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“))
1817ad4ant14 749 . . . . . . . . . 10 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“))
19 sseq2 4003 . . . . . . . . . . . . 13 ((Vtxā€˜š“) = (Vtxā€˜šµ) → (((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“) ↔ ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ)))
2019adantr 480 . . . . . . . . . . . 12 (((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ)) → (((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“) ↔ ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ)))
2120adantl 481 . . . . . . . . . . 11 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“) ↔ ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ)))
2221adantr 480 . . . . . . . . . 10 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → (((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜š“) ↔ ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ)))
2318, 22mpbid 231 . . . . . . . . 9 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → ((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ))
24 resiima 6069 . . . . . . . . 9 (((iEdgā€˜š“)ā€˜š‘–) āŠ† (Vtxā€˜šµ) → (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜š“)ā€˜š‘–))
2523, 24syl 17 . . . . . . . 8 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜š“)ā€˜š‘–))
26 fvresi 7167 . . . . . . . . . 10 (š‘– ∈ dom (iEdgā€˜š“) → (( I ↾ dom (iEdgā€˜š“))ā€˜š‘–) = š‘–)
2726adantl 481 . . . . . . . . 9 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → (( I ↾ dom (iEdgā€˜š“))ā€˜š‘–) = š‘–)
2827fveq2d 6889 . . . . . . . 8 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → ((iEdgā€˜š“)ā€˜(( I ↾ dom (iEdgā€˜š“))ā€˜š‘–)) = ((iEdgā€˜š“)ā€˜š‘–))
29 id 22 . . . . . . . . . . . 12 ((iEdgā€˜š“) = (iEdgā€˜šµ) → (iEdgā€˜š“) = (iEdgā€˜šµ))
30 dmeq 5897 . . . . . . . . . . . . . 14 ((iEdgā€˜š“) = (iEdgā€˜šµ) → dom (iEdgā€˜š“) = dom (iEdgā€˜šµ))
3130reseq2d 5975 . . . . . . . . . . . . 13 ((iEdgā€˜š“) = (iEdgā€˜šµ) → ( I ↾ dom (iEdgā€˜š“)) = ( I ↾ dom (iEdgā€˜šµ)))
3231fveq1d 6887 . . . . . . . . . . . 12 ((iEdgā€˜š“) = (iEdgā€˜šµ) → (( I ↾ dom (iEdgā€˜š“))ā€˜š‘–) = (( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–))
3329, 32fveq12d 6892 . . . . . . . . . . 11 ((iEdgā€˜š“) = (iEdgā€˜šµ) → ((iEdgā€˜š“)ā€˜(( I ↾ dom (iEdgā€˜š“))ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3433adantl 481 . . . . . . . . . 10 (((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ)) → ((iEdgā€˜š“)ā€˜(( I ↾ dom (iEdgā€˜š“))ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3534adantl 481 . . . . . . . . 9 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → ((iEdgā€˜š“)ā€˜(( I ↾ dom (iEdgā€˜š“))ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3635adantr 480 . . . . . . . 8 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → ((iEdgā€˜š“)ā€˜(( I ↾ dom (iEdgā€˜š“))ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3725, 28, 363eqtr2d 2772 . . . . . . 7 ((((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) ∧ š‘– ∈ dom (iEdgā€˜š“)) → (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3837ralrimiva 3140 . . . . . 6 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
3914, 38jca 511 . . . . 5 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–))))
40 f1oeq1 6815 . . . . . 6 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → (š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ↔ ( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ)))
41 fveq1 6884 . . . . . . . . 9 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → (š‘”ā€˜š‘–) = (( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–))
4241fveq2d 6889 . . . . . . . 8 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))
4342eqeq2d 2737 . . . . . . 7 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → ((( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)) ↔ (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–))))
4443ralbidv 3171 . . . . . 6 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → (āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)) ↔ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–))))
4540, 44anbi12d 630 . . . . 5 (š‘” = ( I ↾ dom (iEdgā€˜šµ)) → ((š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))) ↔ (( I ↾ dom (iEdgā€˜šµ)):dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(( I ↾ dom (iEdgā€˜šµ))ā€˜š‘–)))))
469, 39, 45spcedv 3582 . . . 4 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))
476, 46jca 511 . . 3 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (( I ↾ (Vtxā€˜šµ)):(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)))))
48 f1oeq1 6815 . . . 4 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → (š‘“:(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ↔ ( I ↾ (Vtxā€˜šµ)):(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ)))
49 imaeq1 6048 . . . . . . . 8 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → (š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)))
5049eqeq1d 2728 . . . . . . 7 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → ((š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)) ↔ (( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))
5150ralbidv 3171 . . . . . 6 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → (āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)) ↔ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))
5251anbi2d 628 . . . . 5 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → ((š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))) ↔ (š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)))))
5352exbidv 1916 . . . 4 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → (āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))) ↔ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)))))
5448, 53anbi12d 630 . . 3 (š‘“ = ( I ↾ (Vtxā€˜šµ)) → ((š‘“:(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)))) ↔ (( I ↾ (Vtxā€˜šµ)):(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(( I ↾ (Vtxā€˜šµ)) ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))))
552, 47, 54spcedv 3582 . 2 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → āˆƒš‘“(š‘“:(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–)))))
56 eqid 2726 . . . 4 (Vtxā€˜šµ) = (Vtxā€˜šµ)
57 eqid 2726 . . . 4 (iEdgā€˜šµ) = (iEdgā€˜šµ)
5815, 56, 16, 57isomgr 47068 . . 3 ((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) → (š“ IsomGr šµ ↔ āˆƒš‘“(š‘“:(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))))
5958adantr 480 . 2 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → (š“ IsomGr šµ ↔ āˆƒš‘“(š‘“:(Vtxā€˜š“)–1-1-onto→(Vtxā€˜šµ) ∧ āˆƒš‘”(š‘”:dom (iEdgā€˜š“)–1-1-onto→dom (iEdgā€˜šµ) ∧ āˆ€š‘– ∈ dom (iEdgā€˜š“)(š‘“ ā€œ ((iEdgā€˜š“)ā€˜š‘–)) = ((iEdgā€˜šµ)ā€˜(š‘”ā€˜š‘–))))))
6055, 59mpbird 257 1 (((š“ ∈ UHGraph ∧ šµ ∈ š‘Œ) ∧ ((Vtxā€˜š“) = (Vtxā€˜šµ) ∧ (iEdgā€˜š“) = (iEdgā€˜šµ))) → š“ IsomGr šµ)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  āˆƒwex 1773   ∈ wcel 2098  āˆ€wral 3055  Vcvv 3468   āŠ† wss 3943   class class class wbr 5141   I cid 5566  dom cdm 5669   ↾ cres 5671   ā€œ cima 5672  ā€“1-1-onto→wf1o 6536  ā€˜cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  UHGraphcuhgr 28824   IsomGr cisomgr 47064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-uhgr 28826  df-isomgr 47066
This theorem is referenced by:  isomgrref  47080  strisomgrop  47085
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