Theorem isomuspgrlem2c 44069

Description: Lemma 3 for isomuspgrlem2 44072. (Contributed by AV, 29-Nov-2022.)

Hypotheses

Ref	Expression
isomushgr.v	⊢ 𝑉 = (Vtx‘𝐴)
isomushgr.w	⊢ 𝑊 = (Vtx‘𝐵)
isomushgr.e	⊢ 𝐸 = (Edg‘𝐴)
isomushgr.k	⊢ 𝐾 = (Edg‘𝐵)
isomuspgrlem2.g	⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
isomuspgrlem2.a	⊢ (𝜑 → 𝐴 ∈ USPGraph)
isomuspgrlem2.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
isomuspgrlem2.i	⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾))
isomuspgrlem2.x	⊢ (𝜑 → 𝐹 ∈ 𝑋)

Assertion

Ref	Expression
isomuspgrlem2c	⊢ (𝜑 → 𝐺:𝐸–1-1→𝐾)

Distinct variable groups:   𝑎,𝑏,𝑥   𝑥,𝐴   𝑥,𝐵   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊   𝑥,𝐹   𝑥,𝑋   𝐸,𝑎,𝑏   𝐹,𝑎,𝑏   𝐾,𝑎,𝑏   𝑉,𝑎,𝑏   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐺(𝑥,𝑎,𝑏)   𝑊(𝑎,𝑏)   𝑋(𝑎,𝑏)

Proof of Theorem isomuspgrlem2c

Dummy variables 𝑒 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isomushgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐴)
2		isomushgr.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐵)
3		isomushgr.e	. . 3 ⊢ 𝐸 = (Edg‘𝐴)
4		isomushgr.k	. . 3 ⊢ 𝐾 = (Edg‘𝐵)
5		isomuspgrlem2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
6		isomuspgrlem2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ USPGraph)
7		isomuspgrlem2.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
8		isomuspgrlem2.i	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾))
9	1, 2, 3, 4, 5, 6, 7, 8	isomuspgrlem2b 44068	. 2 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)
10		isomuspgrlem2.x	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
11	1, 2, 3, 4, 5	isomuspgrlem2a 44067	. . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))
12	10, 11	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))
13		imaeq2 5918	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑐 → (𝐹 “ 𝑒) = (𝐹 “ 𝑐))
14		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑐 → (𝐺‘𝑒) = (𝐺‘𝑐))
15	13, 14	eqeq12d 2836	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → ((𝐹 “ 𝑒) = (𝐺‘𝑒) ↔ (𝐹 “ 𝑐) = (𝐺‘𝑐)))
16	15	rspcv 3615	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐸 → (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒) → (𝐹 “ 𝑐) = (𝐺‘𝑐)))
17	16	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒) → (𝐹 “ 𝑐) = (𝐺‘𝑐)))
18	17	imp 409	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐹 “ 𝑐) = (𝐺‘𝑐))
19	18	eqcomd 2826	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐺‘𝑐) = (𝐹 “ 𝑐))
20		imaeq2 5918	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑑 → (𝐹 “ 𝑒) = (𝐹 “ 𝑑))
21		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑑 → (𝐺‘𝑒) = (𝐺‘𝑑))
22	20, 21	eqeq12d 2836	. . . . . . . . . 10 ⊢ (𝑒 = 𝑑 → ((𝐹 “ 𝑒) = (𝐺‘𝑒) ↔ (𝐹 “ 𝑑) = (𝐺‘𝑑)))
23	22	rspcv 3615	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝐸 → (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒) → (𝐹 “ 𝑑) = (𝐺‘𝑑)))
24	23	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒) → (𝐹 “ 𝑑) = (𝐺‘𝑑)))
25	24	imp 409	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐹 “ 𝑑) = (𝐺‘𝑑))
26	25	eqcomd 2826	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐺‘𝑑) = (𝐹 “ 𝑑))
27	19, 26	eqeq12d 2836	. . . . 5 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → ((𝐺‘𝑐) = (𝐺‘𝑑) ↔ (𝐹 “ 𝑐) = (𝐹 “ 𝑑)))
28	12, 27	mpidan 687	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → ((𝐺‘𝑐) = (𝐺‘𝑑) ↔ (𝐹 “ 𝑐) = (𝐹 “ 𝑑)))
29		f1of1 6607	. . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
30	7, 29	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑊)
31		uspgrupgr 26959	. . . . . . . 8 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
32		upgruhgr 26885	. . . . . . . . 9 ⊢ (𝐴 ∈ UPGraph → 𝐴 ∈ UHGraph)
33	3	eleq2i 2903	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐸 ↔ 𝑐 ∈ (Edg‘𝐴))
34		edguhgr 26912	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑐 ∈ (Edg‘𝐴)) → 𝑐 ∈ 𝒫 (Vtx‘𝐴))
35		elpwi 4541	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 (Vtx‘𝐴) → 𝑐 ⊆ (Vtx‘𝐴))
36	35, 1	sseqtrrdi 4011	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 (Vtx‘𝐴) → 𝑐 ⊆ 𝑉)
37	34, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑐 ∈ (Edg‘𝐴)) → 𝑐 ⊆ 𝑉)
38	37	ex 415	. . . . . . . . . . 11 ⊢ (𝐴 ∈ UHGraph → (𝑐 ∈ (Edg‘𝐴) → 𝑐 ⊆ 𝑉))
39	33, 38	syl5bi 244	. . . . . . . . . 10 ⊢ (𝐴 ∈ UHGraph → (𝑐 ∈ 𝐸 → 𝑐 ⊆ 𝑉))
40	3	eleq2i 2903	. . . . . . . . . . 11 ⊢ (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Edg‘𝐴))
41		edguhgr 26912	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑑 ∈ (Edg‘𝐴)) → 𝑑 ∈ 𝒫 (Vtx‘𝐴))
42		elpwi 4541	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝒫 (Vtx‘𝐴) → 𝑑 ⊆ (Vtx‘𝐴))
43	42, 1	sseqtrrdi 4011	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝒫 (Vtx‘𝐴) → 𝑑 ⊆ 𝑉)
44	41, 43	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑑 ∈ (Edg‘𝐴)) → 𝑑 ⊆ 𝑉)
45	44	ex 415	. . . . . . . . . . 11 ⊢ (𝐴 ∈ UHGraph → (𝑑 ∈ (Edg‘𝐴) → 𝑑 ⊆ 𝑉))
46	40, 45	syl5bi 244	. . . . . . . . . 10 ⊢ (𝐴 ∈ UHGraph → (𝑑 ∈ 𝐸 → 𝑑 ⊆ 𝑉))
47	39, 46	anim12d 610	. . . . . . . . 9 ⊢ (𝐴 ∈ UHGraph → ((𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐 ⊆ 𝑉 ∧ 𝑑 ⊆ 𝑉)))
48	32, 47	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ UPGraph → ((𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐 ⊆ 𝑉 ∧ 𝑑 ⊆ 𝑉)))
49	6, 31, 48	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐 ⊆ 𝑉 ∧ 𝑑 ⊆ 𝑉)))
50	49	imp 409	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → (𝑐 ⊆ 𝑉 ∧ 𝑑 ⊆ 𝑉))
51		f1imaeq 7016	. . . . . 6 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ (𝑐 ⊆ 𝑉 ∧ 𝑑 ⊆ 𝑉)) → ((𝐹 “ 𝑐) = (𝐹 “ 𝑑) ↔ 𝑐 = 𝑑))
52	30, 50, 51	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → ((𝐹 “ 𝑐) = (𝐹 “ 𝑑) ↔ 𝑐 = 𝑑))
53	52	biimpd 231	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → ((𝐹 “ 𝑐) = (𝐹 “ 𝑑) → 𝑐 = 𝑑))
54	28, 53	sylbid 242	. . 3 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) → ((𝐺‘𝑐) = (𝐺‘𝑑) → 𝑐 = 𝑑))
55	54	ralrimivva 3190	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐸 ∀𝑑 ∈ 𝐸 ((𝐺‘𝑐) = (𝐺‘𝑑) → 𝑐 = 𝑑))
56		dff13 7006	. 2 ⊢ (𝐺:𝐸–1-1→𝐾 ↔ (𝐺:𝐸⟶𝐾 ∧ ∀𝑐 ∈ 𝐸 ∀𝑑 ∈ 𝐸 ((𝐺‘𝑐) = (𝐺‘𝑑) → 𝑐 = 𝑑)))
57	9, 55, 56	sylanbrc 585	1 ⊢ (𝜑 → 𝐺:𝐸–1-1→𝐾)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3137   ⊆ wss 3929  𝒫 cpw 4532  {cpr 4562   ↦ cmpt 5139   “ cima 5551  ⟶wf 6344  –1-1→wf1 6345  –1-1-onto→wf1o 6347  ‘cfv 6348  Vtxcvtx 26779  Edgcedg 26830  UHGraphcuhgr 26839  UPGraphcupgr 26863  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:  isomuspgrlem2e  44071