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Theorem isomgr 44951
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 2738 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑓 = 𝑓)
2 fveq2 6717 . . . . . . 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 2796 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑥) = 𝑉)
6 fveq2 6717 . . . . . . 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 2796 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (Vtx‘𝑦) = 𝑊)
101, 5, 9f1oeq123d 6655 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ↔ 𝑓:𝑉1-1-onto𝑊))
11 eqidd 2738 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑔 = 𝑔)
12 fveq2 6717 . . . . . . . . . 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 2796 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑥) = 𝐼)
1615dmeqd 5774 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → dom (iEdg‘𝑥) = dom 𝐼)
17 fveq2 6717 . . . . . . . . . 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 2796 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (iEdg‘𝑦) = 𝐽)
2120dmeqd 5774 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → dom (iEdg‘𝑦) = dom 𝐽)
2211, 16, 21f1oeq123d 6655 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ↔ 𝑔:dom 𝐼1-1-onto→dom 𝐽))
2315fveq1d 6719 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((iEdg‘𝑥)‘𝑖) = (𝐼𝑖))
2423imaeq2d 5929 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑓 “ ((iEdg‘𝑥)‘𝑖)) = (𝑓 “ (𝐼𝑖)))
2520fveq1d 6719 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ((iEdg‘𝑦)‘(𝑔𝑖)) = (𝐽‘(𝑔𝑖)))
2624, 25eqeq12d 2753 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)) ↔ (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
2716, 26raleqbidv 3313 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
2822, 27anbi12d 634 . . . . 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 1929 . . . 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 634 . . 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 1929 . 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 44949 . 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 5415 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 1543  wex 1787  wcel 2110  wral 3061   class class class wbr 5053  dom cdm 5551  cima 5554  1-1-ontowf1o 6379  cfv 6380  Vtxcvtx 27087  iEdgciedg 27088   IsomGr cisomgr 44947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isomgr 44949
This theorem is referenced by:  isisomgr  44952  isomgreqve  44953  isomushgr  44954  isomgrsym  44964  isomgrtr  44967  ushrisomgr  44969
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