|   | Mathbox for Alexander van der Vekens | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoresfob | Structured version Visualization version GIF version | ||
| Description: A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-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 | 
|---|---|
| fcoresfob | ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fcores.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐹:𝐴⟶𝐵) | 
| 3 | fcores.e | . . 3 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
| 4 | fcores.p | . . 3 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
| 5 | fcores.x | . . 3 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
| 6 | fcores.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝐺:𝐶⟶𝐷) | 
| 8 | fcores.y | . . 3 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
| 9 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷) | |
| 10 | 2, 3, 4, 5, 7, 8, 9 | fcoresfo 47083 | . 2 ⊢ ((𝜑 ∧ (𝐺 ∘ 𝐹):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷) | 
| 11 | 1, 3, 4, 5 | fcoreslem3 47077 | . . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) | 
| 12 | 11 | anim1ci 616 | . . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸)) | 
| 13 | foco 6834 | . . . 4 ⊢ ((𝑌:𝐸–onto→𝐷 ∧ 𝑋:𝑃–onto→𝐸) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷) | |
| 14 | 12, 13 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝑌 ∘ 𝑋):𝑃–onto→𝐷) | 
| 15 | 1, 3, 4, 5, 6, 8 | fcores 47079 | . . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) | 
| 16 | 15 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) | 
| 17 | foeq1 6816 | . . . 4 ⊢ ((𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷)) | |
| 18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ (𝑌 ∘ 𝑋):𝑃–onto→𝐷)) | 
| 19 | 14, 18 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑌:𝐸–onto→𝐷) → (𝐺 ∘ 𝐹):𝑃–onto→𝐷) | 
| 20 | 10, 19 | impbida 801 | 1 ⊢ (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∩ cin 3950 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 ∘ ccom 5689 ⟶wf 6557 –onto→wfo 6559 | 
| 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: fcoresf1ob 47085 | 
| Copyright terms: Public domain | W3C validator |