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Mirrors > Home > MPE Home > Th. List > Mathboxes > isomuspgrlem2a | Structured version Visualization version GIF version |
Description: Lemma 1 for isomuspgrlem2 44005. (Contributed by AV, 29-Nov-2022.) |
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)) |
Ref | Expression |
---|---|
isomuspgrlem2a | ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) |
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 5927 | . . . . 5 ⊢ (𝑥 = 𝑒 → (𝐹 “ 𝑥) = (𝐹 “ 𝑒)) | |
4 | 3 | adantl 484 | . . . 4 ⊢ (((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) ∧ 𝑥 = 𝑒) → (𝐹 “ 𝑥) = (𝐹 “ 𝑒)) |
5 | simpr 487 | . . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐸) | |
6 | imaexg 7622 | . . . . 5 ⊢ (𝐹 ∈ 𝑋 → (𝐹 “ 𝑒) ∈ V) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) ∈ V) |
8 | 2, 4, 5, 7 | fvmptd 6777 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐺‘𝑒) = (𝐹 “ 𝑒)) |
9 | 8 | eqcomd 2829 | . 2 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) = (𝐺‘𝑒)) |
10 | 9 | ralrimiva 3184 | 1 ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ↦ cmpt 5148 “ cima 5560 ‘cfv 6357 Vtxcvtx 26783 Edgcedg 26834 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: isomuspgrlem2c 44002 isomuspgrlem2d 44003 isomuspgrlem2 44005 |
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