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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 4246 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
5 | 3, 4 | eqsstrdi 4050 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) |
6 | 1, 5 | fssresd 6776 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷) |
7 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
8 | 7 | feq1i 6728 | . . 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 47015 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
14 | fof 6821 | . . 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 47017 | . . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) |
18 | 17 | eqcomd 2741 | . . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹)) |
19 | foeq1 6817 | . . . 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 7129 | . 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷) | |
23 | 9, 15, 21, 22 | syl3anc 1370 | 1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∩ cin 3962 ◡ccnv 5688 ran crn 5690 ↾ cres 5691 “ cima 5692 ∘ ccom 5693 ⟶wf 6559 –onto→wfo 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 |
This theorem is referenced by: fcoresfob 47022 funfocofob 47028 |
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