Theorem isushgrm 15880

Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Hypotheses

Ref	Expression
isuhgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isuhgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
isushgrm	⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))

Distinct variable groups:   𝑗,𝑠   𝑉,𝑠

Allowed substitution hints:   𝑈(𝑗,𝑠)   𝐸(𝑗,𝑠)   𝐺(𝑗,𝑠)   𝑉(𝑗)

Proof of Theorem isushgrm

Dummy variables 𝑔 ℎ 𝑣 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ushgrm 15878	. . 3 ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
2	1	eleq2i 2296	. 2 ⊢ (𝐺 ∈ USHGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}})
3		fveq2 5629	. . . . 5 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = (iEdg‘𝐺))
4		isuhgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
5	3, 4	eqtr4di 2280	. . . 4 ⊢ (ℎ = 𝐺 → (iEdg‘ℎ) = 𝐸)
6	3	dmeqd 4925	. . . . 5 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom (iEdg‘𝐺))
7	4	eqcomi 2233	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐸
8	7	dmeqi 4924	. . . . 5 ⊢ dom (iEdg‘𝐺) = dom 𝐸
9	6, 8	eqtrdi 2278	. . . 4 ⊢ (ℎ = 𝐺 → dom (iEdg‘ℎ) = dom 𝐸)
10		fveq2 5629	. . . . . . 7 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
11		isuhgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	eqtr4di 2280	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
13	12	pweqd 3654	. . . . 5 ⊢ (ℎ = 𝐺 → 𝒫 (Vtx‘ℎ) = 𝒫 𝑉)
14	13	rabeqdv 2793	. . . 4 ⊢ (ℎ = 𝐺 → {𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠})
15	5, 9, 14	f1eq123d 5566	. . 3 ⊢ (ℎ = 𝐺 → ((iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ 𝐸:dom 𝐸–1-1→{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
16		vtxex 15827	. . . . . . 7 ⊢ (𝑔 ∈ V → (Vtx‘𝑔) ∈ V)
17	16	elv 2803	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
18	17	a1i 9	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) ∈ V)
19		fveq2 5629	. . . . 5 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
20		iedgex 15828	. . . . . . . 8 ⊢ (𝑔 ∈ V → (iEdg‘𝑔) ∈ V)
21	20	elv 2803	. . . . . . 7 ⊢ (iEdg‘𝑔) ∈ V
22	21	a1i 9	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) ∈ V)
23		fveq2 5629	. . . . . . 7 ⊢ (𝑔 = ℎ → (iEdg‘𝑔) = (iEdg‘ℎ))
24	23	adantr 276	. . . . . 6 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → (iEdg‘𝑔) = (iEdg‘ℎ))
25		simpr 110	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → 𝑒 = (iEdg‘ℎ))
26	25	dmeqd 4925	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → dom 𝑒 = dom (iEdg‘ℎ))
27		simpr 110	. . . . . . . . . 10 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝑣 = (Vtx‘ℎ))
28	27	pweqd 3654	. . . . . . . . 9 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → 𝒫 𝑣 = 𝒫 (Vtx‘ℎ))
29	28	rabeqdv 2793	. . . . . . . 8 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → {𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠})
30	29	adantr 276	. . . . . . 7 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → {𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠})
31	25, 26, 30	f1eq123d 5566	. . . . . 6 ⊢ (((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) ∧ 𝑒 = (iEdg‘ℎ)) → (𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
32	22, 24, 31	sbcied2 3066	. . . . 5 ⊢ ((𝑔 = ℎ ∧ 𝑣 = (Vtx‘ℎ)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
33	18, 19, 32	sbcied2 3066	. . . 4 ⊢ (𝑔 = ℎ → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠}))
34	33	cbvabv 2354	. . 3 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}} = {ℎ ∣ (iEdg‘ℎ):dom (iEdg‘ℎ)–1-1→{𝑠 ∈ 𝒫 (Vtx‘ℎ) ∣ ∃𝑗 𝑗 ∈ 𝑠}}
35	15, 34	elab2g 2950	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}} ↔ 𝐸:dom 𝐸–1-1→{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))
36	2, 35	bitrid 192	1 ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→{𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  {crab 2512  Vcvv 2799  [wsbc 3028  𝒫 cpw 3649  dom cdm 4719  –1-1→wf1 5315  ‘cfv 5318  Vtxcvtx 15821  iEdgciedg 15822  USHGraphcushgr 15876

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8327  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-dec 9587  df-ndx 13043  df-slot 13044  df-base 13046  df-edgf 15814  df-vtx 15823  df-iedg 15824  df-ushgrm 15878

This theorem is referenced by:  ushgrfm  15882  uspgrushgr  15986