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Mirrors > Home > MPE Home > Th. List > usgredg2vtxeuALT | Structured version Visualization version GIF version |
Description: Alternate proof of usgredg2vtxeu 26997, using edgiedgb 26833, the general translation from (iEdg‘𝐺) to (Edg‘𝐺). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
usgredg2vtxeuALT | ⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruhgr 26962 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
2 | eqid 2821 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 2 | uhgredgiedgb 26905 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥))) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥))) |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
6 | 5, 2 | usgredgreu 26994 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝑌 ∈ ((iEdg‘𝐺)‘𝑥)) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}) |
7 | 6 | 3expia 1117 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})) |
8 | 7 | 3adant3 1128 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})) |
9 | eleq2 2901 | . . . . . . . 8 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 ↔ 𝑌 ∈ ((iEdg‘𝐺)‘𝑥))) | |
10 | eqeq1 2825 | . . . . . . . . 9 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝐸 = {𝑌, 𝑦} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})) | |
11 | 10 | reubidv 3390 | . . . . . . . 8 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦} ↔ ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})) |
12 | 9, 11 | imbi12d 347 | . . . . . . 7 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → ((𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) ↔ (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))) |
13 | 12 | 3ad2ant3 1131 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → ((𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) ↔ (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))) |
14 | 8, 13 | mpbird 259 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})) |
15 | 14 | 3exp 1115 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})))) |
16 | 15 | rexlimdv 3283 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}))) |
17 | 4, 16 | sylbid 242 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ (Edg‘𝐺) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}))) |
18 | 17 | 3imp 1107 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ∃!wreu 3140 {cpr 4563 dom cdm 5550 ‘cfv 6350 Vtxcvtx 26775 iEdgciedg 26776 Edgcedg 26826 UHGraphcuhgr 26835 USGraphcusgr 26928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-edg 26827 df-uhgr 26837 df-upgr 26861 df-umgr 26862 df-uspgr 26929 df-usgr 26930 |
This theorem is referenced by: (None) |
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