Theorem grstructd2dom 15972

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 15953	. . . . 5 ⊢ ((𝑆 ∈ 𝑋 ∧ Fun (𝑆 ∖ {∅}) ∧ 2o ≼ dom 𝑆) → (Vtx‘𝑆) = (Base‘𝑆))
6	1, 3, 4, 5	syl3anc 1274	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆))
7		grstructd.b	. . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉)
8	6, 7	eqtrd 2264	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
9		funiedgdm2domval 15954	. . . . 5 ⊢ ((𝑆 ∈ 𝑋 ∧ Fun (𝑆 ∖ {∅}) ∧ 2o ≼ dom 𝑆) → (iEdg‘𝑆) = (.ef‘𝑆))
10	1, 3, 4, 9	syl3anc 1274	. . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆))
11		grstructd.e	. . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸)
12	10, 11	eqtrd 2264	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸)
13	8, 12	jca 306	. 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))
14		nfcv 2375	. . 3 ⊢ Ⅎ𝑔𝑆
15		nfv 1577	. . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)
16		nfsbc1v 3051	. . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓
17	15, 16	nfim 1621	. . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓)
18		fveqeq2 5657	. . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉))
19		fveqeq2 5657	. . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸))
20	18, 19	anbi12d 473	. . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)))
21		sbceq1a 3042	. . . 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 2889	. 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 1396   = wceq 1398   ∈ wcel 2202  [wsbc 3032   ∖ cdif 3198  ∅c0 3496  {csn 3673   class class class wbr 4093  dom cdm 4731  Fun wfun 5327  ‘cfv 5333  2_oc2o 6619   ≼ cdom 6951  Basecbs 13145  .efcedgf 15928  Vtxcvtx 15936  iEdgciedg 15937

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-dom 6954  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939

This theorem is referenced by:  grstructeld2dom  15974