Theorem isomuspgrlem2 44018

Description: Lemma 2 for isomuspgr 44019. (Contributed by AV, 1-Dec-2022.)

Hypotheses

Ref	Expression
isomushgr.v	⊢ 𝑉 = (Vtx‘𝐴)
isomushgr.w	⊢ 𝑊 = (Vtx‘𝐵)
isomushgr.e	⊢ 𝐸 = (Edg‘𝐴)
isomushgr.k	⊢ 𝐾 = (Edg‘𝐵)

Assertion

Ref	Expression
isomuspgrlem2	⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))

Distinct variable groups:   𝐴,𝑒,𝑓,𝑔   𝐵,𝑒,𝑓,𝑔   𝑒,𝐸,𝑔   𝑔,𝐾   𝑒,𝑉,𝑔   𝑒,𝑊,𝑔   𝑎,𝑏,𝑔,𝑓   𝐸,𝑎,𝑏   𝐾,𝑎,𝑏   𝑉,𝑎,𝑏

Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐸(𝑓)   𝐾(𝑒,𝑓)   𝑉(𝑓)   𝑊(𝑓,𝑎,𝑏)

Proof of Theorem isomuspgrlem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomushgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐴)
2	1	fvexi 6684	. . . 4 ⊢ 𝐸 ∈ V
3	2	mptex 6986	. . 3 ⊢ (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) ∈ V
4		isomushgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐴)
5		isomushgr.w	. . . . 5 ⊢ 𝑊 = (Vtx‘𝐵)
6		isomushgr.k	. . . . 5 ⊢ 𝐾 = (Edg‘𝐵)
7		eqid 2821	. . . . 5 ⊢ (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))
8		simplll 773	. . . . 5 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → 𝐴 ∈ USPGraph)
9		simplr 767	. . . . 5 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → 𝑓:𝑉–1-1-onto→𝑊)
10		simpr 487	. . . . 5 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))
11		vex 3497	. . . . . 6 ⊢ 𝑓 ∈ V
12	11	a1i 11	. . . . 5 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → 𝑓 ∈ V)
13		simpllr 774	. . . . 5 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → 𝐵 ∈ USPGraph)
14	4, 5, 1, 6, 7, 8, 9, 10, 12, 13	isomuspgrlem2e 44017	. . . 4 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾)
15	4, 5, 1, 6, 7	isomuspgrlem2a 44013	. . . . 5 ⊢ (𝑓 ∈ V → ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒))
16	11, 15	mp1i 13	. . . 4 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒))
17	14, 16	jca 514	. . 3 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒)))
18		f1oeq1 6604	. . . . 5 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → (𝑔:𝐸–1-1-onto→𝐾 ↔ (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾))
19		fveq1 6669	. . . . . . 7 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → (𝑔‘𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒))
20	19	eqeq2d 2832	. . . . . 6 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒)))
21	20	ralbidv 3197	. . . . 5 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → (∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒)))
22	18, 21	anbi12d 632	. . . 4 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → ((𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)) ↔ ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒))))
23	22	spcegv 3597	. . 3 ⊢ ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) ∈ V → (((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒)) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))
24	3, 17, 23	mpsyl 68	. 2 ⊢ ((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))
25	24	ex 415	1 ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  {cpr 4569   ↦ cmpt 5146   “ cima 5558  –1-1-onto→wf1o 6354  ‘cfv 6355  Vtxcvtx 26781  Edgcedg 26832  USPGraphcuspgr 26933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3496  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 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-fz 12894  df-hash 13692  df-edg 26833  df-uhgr 26843  df-upgr 26867  df-uspgr 26935

This theorem is referenced by:  isomuspgr  44019