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Theorem isomgr 44280
Description: The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.)
Hypotheses
Ref Expression
isomgr.v 𝑉 = (Vtx‘𝐴)
isomgr.w 𝑊 = (Vtx‘𝐵)
isomgr.i 𝐼 = (iEdg‘𝐴)
isomgr.j 𝐽 = (iEdg‘𝐵)
Assertion
Ref Expression
isomgr ((𝐴𝑋𝐵𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Distinct variable groups:   𝐴,𝑓,𝑔,𝑖   𝐵,𝑓,𝑔,𝑖   𝑖,𝐼
Allowed substitution hints:   𝐼(𝑓,𝑔)   𝐽(𝑓,𝑔,𝑖)   𝑉(𝑓,𝑔,𝑖)   𝑊(𝑓,𝑔,𝑖)   𝑋(𝑓,𝑔,𝑖)   𝑌(𝑓,𝑔,𝑖)

Proof of Theorem isomgr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2823 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑓 = 𝑓)
2 fveq2 6652 . . . . . . 7 (𝑥 = 𝐴 → (Vtx‘𝑥) = (Vtx‘𝐴))
32adantr 484 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑥) = (Vtx‘𝐴))
4 isomgr.v . . . . . 6 𝑉 = (Vtx‘𝐴)
53, 4eqtr4di 2875 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑥) = 𝑉)
6 fveq2 6652 . . . . . . 7 (𝑦 = 𝐵 → (Vtx‘𝑦) = (Vtx‘𝐵))
76adantl 485 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑦) = (Vtx‘𝐵))
8 isomgr.w . . . . . 6 𝑊 = (Vtx‘𝐵)
97, 8eqtr4di 2875 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑦) = 𝑊)
101, 5, 9f1oeq123d 6592 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ↔ 𝑓:𝑉1-1-onto𝑊))
11 eqidd 2823 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑔 = 𝑔)
12 fveq2 6652 . . . . . . . . . 10 (𝑥 = 𝐴 → (iEdg‘𝑥) = (iEdg‘𝐴))
1312adantr 484 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑥) = (iEdg‘𝐴))
14 isomgr.i . . . . . . . . 9 𝐼 = (iEdg‘𝐴)
1513, 14eqtr4di 2875 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑥) = 𝐼)
1615dmeqd 5751 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → dom (iEdg‘𝑥) = dom 𝐼)
17 fveq2 6652 . . . . . . . . . 10 (𝑦 = 𝐵 → (iEdg‘𝑦) = (iEdg‘𝐵))
1817adantl 485 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑦) = (iEdg‘𝐵))
19 isomgr.j . . . . . . . . 9 𝐽 = (iEdg‘𝐵)
2018, 19eqtr4di 2875 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑦) = 𝐽)
2120dmeqd 5751 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → dom (iEdg‘𝑦) = dom 𝐽)
2211, 16, 21f1oeq123d 6592 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ↔ 𝑔:dom 𝐼1-1-onto→dom 𝐽))
2315fveq1d 6654 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((iEdg‘𝑥)‘𝑖) = (𝐼𝑖))
2423imaeq2d 5907 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑓 “ ((iEdg‘𝑥)‘𝑖)) = (𝑓 “ (𝐼𝑖)))
2520fveq1d 6654 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ((iEdg‘𝑦)‘(𝑔𝑖)) = (𝐽‘(𝑔𝑖)))
2624, 25eqeq12d 2838 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)) ↔ (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
2716, 26raleqbidv 3382 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
2822, 27anbi12d 633 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖))) ↔ (𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))
2928exbidv 1922 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖))) ↔ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))
3010, 29anbi12d 633 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)))) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
3130exbidv 1922 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)))) ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
32 df-isomgr 44278 . 2 IsomGr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖))))}
3331, 32brabga 5398 1 ((𝐴𝑋𝐵𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2114  wral 3130   class class class wbr 5042  dom cdm 5532  cima 5535  1-1-ontowf1o 6333  cfv 6334  Vtxcvtx 26787  iEdgciedg 26788   IsomGr cisomgr 44276
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isomgr 44278
This theorem is referenced by:  isisomgr  44281  isomgreqve  44282  isomushgr  44283  isomgrsym  44293  isomgrtr  44296  ushrisomgr  44298
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