Theorem isomuspgrlem2 44072

Description: Lemma 2 for isomuspgr 44073. (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 6677	. . . 4 ⊢ 𝐸 ∈ V
3	2	mptex 6979	. . 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 2820	. . . . 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 3494	. . . . . 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 44071	. . . 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 44067	. . . . 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 6597	. . . . 5 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → (𝑔:𝐸–1-1-onto→𝐾 ↔ (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)):𝐸–1-1-onto→𝐾))
19		fveq1 6662	. . . . . . 7 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → (𝑔‘𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒))
20	19	eqeq2d 2831	. . . . . 6 ⊢ (𝑔 = (𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥)) → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝑒) = ((𝑥 ∈ 𝐸 ↦ (𝑓 “ 𝑥))‘𝑒)))
21	20	ralbidv 3196	. . . . 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 3594	. . 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 1536  ∃wex 1779   ∈ wcel 2113  ∀wral 3137  Vcvv 3491  {cpr 4562   ↦ cmpt 5139   “ cima 5551  –1-1-onto→wf1o 6347  ‘cfv 6348  Vtxcvtx 26779  Edgcedg 26830  USPGraphcuspgr 26931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-1st 7682  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-2o 8096  df-oadd 8099  df-er 8282  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-dju 9323  df-card 9361  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11632  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12890  df-hash 13688  df-edg 26831  df-uhgr 26841  df-upgr 26865  df-uspgr 26933

This theorem is referenced by:  isomuspgr  44073