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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoresf1lem | Structured version Visualization version GIF version | ||
| Description: Lemma for fcoresf1 47081. (Contributed by AV, 18-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| fcores.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| fcores.e | ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | 
| fcores.p | ⊢ 𝑃 = (◡𝐹 “ 𝐶) | 
| fcores.x | ⊢ 𝑋 = (𝐹 ↾ 𝑃) | 
| fcores.g | ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | 
| fcores.y | ⊢ 𝑌 = (𝐺 ↾ 𝐸) | 
| Ref | Expression | 
|---|---|
| fcoresf1lem | ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fcores.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 2 | fcores.e | . . . . 5 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
| 3 | fcores.p | . . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
| 4 | fcores.x | . . . . 5 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
| 5 | fcores.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
| 6 | fcores.y | . . . . 5 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
| 7 | 1, 2, 3, 4, 5, 6 | fcores 47079 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) | 
| 8 | 7 | fveq1d 6908 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍)) | 
| 9 | 8 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍)) | 
| 10 | 1, 2, 3, 4 | fcoreslem3 47077 | . . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) | 
| 11 | fof 6820 | . . . . 5 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸) | |
| 12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸) | 
| 13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑋:𝑃⟶𝐸) | 
| 14 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃) | |
| 15 | 13, 14 | fvco3d 7009 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑍) = (𝑌‘(𝑋‘𝑍))) | 
| 16 | 9, 15 | eqtrd 2777 | 1 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 | 
| This theorem is referenced by: fcoresf1 47081 | 
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