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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isisomgr | Structured version Visualization version GIF version |
Description: Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) |
Ref | Expression |
---|---|
isomgr.v | ⊢ 𝑉 = (Vtx‘𝐴) |
isomgr.w | ⊢ 𝑊 = (Vtx‘𝐵) |
isomgr.i | ⊢ 𝐼 = (iEdg‘𝐴) |
isomgr.j | ⊢ 𝐽 = (iEdg‘𝐵) |
Ref | Expression |
---|---|
isisomgr | ⊢ (𝐴 IsomGr 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isomgrrel 46490 | . . . 4 ⊢ Rel IsomGr | |
2 | 1 | brrelex12i 5732 | . . 3 ⊢ (𝐴 IsomGr 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | isomgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐴) | |
4 | isomgr.w | . . . 4 ⊢ 𝑊 = (Vtx‘𝐵) | |
5 | isomgr.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐴) | |
6 | isomgr.j | . . . 4 ⊢ 𝐽 = (iEdg‘𝐵) | |
7 | 3, 4, 5, 6 | isomgr 46491 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐴 IsomGr 𝐵 → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) |
9 | 8 | ibi 267 | 1 ⊢ (𝐴 IsomGr 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 class class class wbr 5149 dom cdm 5677 “ cima 5680 –1-1-onto→wf1o 6543 ‘cfv 6544 Vtxcvtx 28256 iEdgciedg 28257 IsomGr cisomgr 46487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isomgr 46489 |
This theorem is referenced by: (None) |
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