Theorem isomuspgrlem2c 43989

Description: Lemma 3 for isomuspgrlem2 43992. (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 43988	. 2 ⊢ (𝜑 → 𝐺:𝐸⟶𝐾)
10		isomuspgrlem2.x	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
11	1, 2, 3, 4, 5	isomuspgrlem2a 43987	. . . . . 6 ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))
12	10, 11	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))
13		imaeq2 5919	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑐 → (𝐹 “ 𝑒) = (𝐹 “ 𝑐))
14		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑐 → (𝐺‘𝑒) = (𝐺‘𝑐))
15	13, 14	eqeq12d 2837	. . . . . . . . . 10 ⊢ (𝑒 = 𝑐 → ((𝐹 “ 𝑒) = (𝐺‘𝑒) ↔ (𝐹 “ 𝑐) = (𝐺‘𝑐)))
16	15	rspcv 3617	. . . . . . . . 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 2827	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐺‘𝑐) = (𝐹 “ 𝑐))
20		imaeq2 5919	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑑 → (𝐹 “ 𝑒) = (𝐹 “ 𝑑))
21		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑑 → (𝐺‘𝑒) = (𝐺‘𝑑))
22	20, 21	eqeq12d 2837	. . . . . . . . . 10 ⊢ (𝑒 = 𝑑 → ((𝐹 “ 𝑒) = (𝐺‘𝑒) ↔ (𝐹 “ 𝑑) = (𝐺‘𝑑)))
23	22	rspcv 3617	. . . . . . . . 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 2827	. . . . . 6 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸)) ∧ ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) → (𝐺‘𝑑) = (𝐹 “ 𝑑))
27	19, 26	eqeq12d 2837	. . . . 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 6608	. . . . . . 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 26955	. . . . . . . 8 ⊢ (𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph)
32		upgruhgr 26881	. . . . . . . . 9 ⊢ (𝐴 ∈ UPGraph → 𝐴 ∈ UHGraph)
33	3	eleq2i 2904	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐸 ↔ 𝑐 ∈ (Edg‘𝐴))
34		edguhgr 26908	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑐 ∈ (Edg‘𝐴)) → 𝑐 ∈ 𝒫 (Vtx‘𝐴))
35		elpwi 4550	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 (Vtx‘𝐴) → 𝑐 ⊆ (Vtx‘𝐴))
36	35, 1	sseqtrrdi 4017	. . . . . . . . . . . . 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 2904	. . . . . . . . . . 11 ⊢ (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Edg‘𝐴))
41		edguhgr 26908	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑑 ∈ (Edg‘𝐴)) → 𝑑 ∈ 𝒫 (Vtx‘𝐴))
42		elpwi 4550	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝒫 (Vtx‘𝐴) → 𝑑 ⊆ (Vtx‘𝐴))
43	42, 1	sseqtrrdi 4017	. . . . . . . . . . . . 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 7017	. . . . . 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 3191	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐸 ∀𝑑 ∈ 𝐸 ((𝐺‘𝑐) = (𝐺‘𝑑) → 𝑐 = 𝑑))
56		dff13 7007	. 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 1533   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3935  𝒫 cpw 4538  {cpr 4562   ↦ cmpt 5138   “ cima 5552  ⟶wf 6345  –1-1→wf1 6346  –1-1-onto→wf1o 6348  ‘cfv 6349  Vtxcvtx 26775  Edgcedg 26826  UHGraphcuhgr 26835  UPGraphcupgr 26859  USPGraphcuspgr 26927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12887  df-hash 13685  df-edg 26827  df-uhgr 26837  df-upgr 26861  df-uspgr 26929

This theorem is referenced by:  isomuspgrlem2e  43991