Theorem grstructd2dom 16203

Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)

Hypotheses

Ref	Expression
gropd.g	⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
gropd.v	⊢ (𝜑 → 𝑉 ∈ 𝑈)
gropd.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)
grstructd.s	⊢ (𝜑 → 𝑆 ∈ 𝑋)
grstructd.f	⊢ (𝜑 → Fun (𝑆 ∖ {∅}))
grstructd2dom.d	⊢ (𝜑 → 2o ≼ dom 𝑆)
grstructd.b	⊢ (𝜑 → (Base‘𝑆) = 𝑉)
grstructd.e	⊢ (𝜑 → (.ef‘𝑆) = 𝐸)

Assertion

Ref	Expression
grstructd2dom	⊢ (𝜑 → [𝑆 / 𝑔]𝜓)

Distinct variable groups:   𝑔,𝐸   𝑔,𝑉   𝜑,𝑔   𝑆,𝑔

Allowed substitution hints:   𝜓(𝑔)   𝑈(𝑔)   𝑊(𝑔)   𝑋(𝑔)

Proof of Theorem grstructd2dom

Step	Hyp	Ref	Expression
1		grstructd.s	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
2		gropd.g	. 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
3		grstructd.f	. . . . 5 ⊢ (𝜑 → Fun (𝑆 ∖ {∅}))
4		grstructd2dom.d	. . . . 5 ⊢ (𝜑 → 2o ≼ dom 𝑆)
5		funvtxdm2domval 16184	. . . . 5 ⊢ ((𝑆 ∈ 𝑋 ∧ Fun (𝑆 ∖ {∅}) ∧ 2o ≼ dom 𝑆) → (Vtx‘𝑆) = (Base‘𝑆))
6	1, 3, 4, 5	syl3anc 1278	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆))
7		grstructd.b	. . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉)
8	6, 7	eqtrd 2271	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
9		funiedgdm2domval 16185	. . . . 5 ⊢ ((𝑆 ∈ 𝑋 ∧ Fun (𝑆 ∖ {∅}) ∧ 2o ≼ dom 𝑆) → (iEdg‘𝑆) = (.ef‘𝑆))
10	1, 3, 4, 9	syl3anc 1278	. . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆))
11		grstructd.e	. . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)
12	10, 11	eqtrd 2271	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸)
13	8, 12	jca 306	. 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))
14		nfcv 2392	. . 3 ⊢ Ⅎ𝑔𝑆
15		nfv 1581	. . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)
16		nfsbc1v 3070	. . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓
17	15, 16	nfim 1625	. . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓)
18		fveqeq2 5699	. . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉))
19		fveqeq2 5699	. . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸))
20	18, 19	anbi12d 477	. . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)))
21		sbceq1a 3061	. . . 4 ⊢ (𝑔 = 𝑆 → (𝜓 ↔ [𝑆 / 𝑔]𝜓))
22	20, 21	imbi12d 234	. . 3 ⊢ (𝑔 = 𝑆 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓)))
23	14, 17, 22	spcgf 2907	. 2 ⊢ (𝑆 ∈ 𝑋 → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓)))
24	1, 2, 13, 23	syl3c 63	1 ⊢ (𝜑 → [𝑆 / 𝑔]𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402   ∈ wcel 2209  [wsbc 3051   ∖ cdif 3217  ∅c0 3520  {csn 3705   class class class wbr 4125  dom cdm 4769  Fun wfun 5366  ‘cfv 5372  2_oc2o 6671   ≼ cdom 7011  Basecbs 13330  .efcedgf 16159  Vtxcvtx 16167  iEdgciedg 16168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-dom 7014  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170

This theorem is referenced by:  grstructeld2dom  16205