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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 46100 | . . . 4 ⊢ Rel IsomGr | |
2 | 1 | brrelex12i 5688 | . . 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 46101 | . . 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 3061 Vcvv 3444 class class class wbr 5106 dom cdm 5634 “ cima 5637 –1-1-onto→wf1o 6496 ‘cfv 6497 Vtxcvtx 27989 iEdgciedg 27990 IsomGr cisomgr 46097 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isomgr 46099 |
This theorem is referenced by: (None) |
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