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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoresfo | Structured version Visualization version GIF version | ||
| Description: If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-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) | 
| fcoresfo.s | ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷) | 
| Ref | Expression | 
|---|---|
| fcoresfo | ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fcores.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
| 2 | fcores.e | . . . . . 6 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
| 3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐶)) | 
| 4 | inss2 4237 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
| 5 | 3, 4 | eqsstrdi 4027 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) | 
| 6 | 1, 5 | fssresd 6774 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷) | 
| 7 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
| 8 | 7 | feq1i 6726 | . . 3 ⊢ (𝑌:𝐸⟶𝐷 ↔ (𝐺 ↾ 𝐸):𝐸⟶𝐷) | 
| 9 | 6, 8 | sylibr 234 | . 2 ⊢ (𝜑 → 𝑌:𝐸⟶𝐷) | 
| 10 | fcores.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 11 | fcores.p | . . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
| 12 | fcores.x | . . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
| 13 | 10, 2, 11, 12 | fcoreslem3 47082 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) | 
| 14 | fof 6819 | . . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸) | |
| 15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸) | 
| 16 | fcoresfo.s | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷) | |
| 17 | 10, 2, 11, 12, 1, 7 | fcores 47084 | . . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) | 
| 18 | 17 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹)) | 
| 19 | foeq1 6815 | . . . 4 ⊢ ((𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹) → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷)) | |
| 20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑌 ∘ 𝑋):𝑃–onto→𝐷 ↔ (𝐺 ∘ 𝐹):𝑃–onto→𝐷)) | 
| 21 | 16, 20 | mpbird 257 | . 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋):𝑃–onto→𝐷) | 
| 22 | foco2 7128 | . 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷) | |
| 23 | 9, 15, 21, 22 | syl3anc 1372 | 1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∩ cin 3949 ◡ccnv 5683 ran crn 5685 ↾ cres 5686 “ cima 5687 ∘ ccom 5688 ⟶wf 6556 –onto→wfo 6558 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 | 
| This theorem is referenced by: fcoresfob 47089 funfocofob 47095 | 
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