Theorem isomuspgrlem2a 44346

Description: Lemma 1 for isomuspgrlem2 44351. (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))

Assertion

Ref	Expression
isomuspgrlem2a	⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))

Distinct variable groups:   𝐴,𝑒   𝐵,𝑒   𝑒,𝐸   𝑒,𝑉   𝑒,𝑊,𝑥   𝑥,𝐴   𝑥,𝐵   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉   𝑥,𝑊   𝑒,𝐹,𝑥   𝑒,𝑋,𝑥

Allowed substitution hints:   𝐺(𝑥,𝑒)   𝐾(𝑒)

Proof of Theorem isomuspgrlem2a

Step	Hyp	Ref	Expression
1		isomuspgrlem2.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
2	1	a1i 11	. . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)))
3		imaeq2 5892	. . . . 5 ⊢ (𝑥 = 𝑒 → (𝐹 “ 𝑥) = (𝐹 “ 𝑒))
4	3	adantl 485	. . . 4 ⊢ (((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) ∧ 𝑥 = 𝑒) → (𝐹 “ 𝑥) = (𝐹 “ 𝑒))
5		simpr 488	. . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸)
6		imaexg 7602	. . . . 5 ⊢ (𝐹 ∈ 𝑋 → (𝐹 “ 𝑒) ∈ V)
7	6	adantr 484	. . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) ∈ V)
8	2, 4, 5, 7	fvmptd 6752	. . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐺‘𝑒) = (𝐹 “ 𝑒))
9	8	eqcomd 2804	. 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) = (𝐺‘𝑒))
10	9	ralrimiva 3149	1 ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ↦ cmpt 5110   “ cima 5522  ‘cfv 6324  Vtxcvtx 26789  Edgcedg 26840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332

This theorem is referenced by:  isomuspgrlem2c  44348  isomuspgrlem2d  44349  isomuspgrlem2  44351