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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 4232 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
5 | 3, 4 | eqsstrdi 4036 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) |
6 | 1, 5 | fssresd 6769 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸):𝐸⟶𝐷) |
7 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
8 | 7 | feq1i 6718 | . . 3 ⊢ (𝑌:𝐸⟶𝐷 ↔ (𝐺 ↾ 𝐸):𝐸⟶𝐷) |
9 | 6, 8 | sylibr 233 | . 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 46476 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
14 | fof 6816 | . . 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 46478 | . . . . 5 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) |
18 | 17 | eqcomd 2734 | . . . 4 ⊢ (𝜑 → (𝑌 ∘ 𝑋) = (𝐺 ∘ 𝐹)) |
19 | foeq1 6812 | . . . 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 256 | . 2 ⊢ (𝜑 → (𝑌 ∘ 𝑋):𝑃–onto→𝐷) |
22 | foco2 7124 | . 2 ⊢ ((𝑌:𝐸⟶𝐷 ∧ 𝑋:𝑃⟶𝐸 ∧ (𝑌 ∘ 𝑋):𝑃–onto→𝐷) → 𝑌:𝐸–onto→𝐷) | |
23 | 9, 15, 21, 22 | syl3anc 1368 | 1 ⊢ (𝜑 → 𝑌:𝐸–onto→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∩ cin 3948 ◡ccnv 5681 ran crn 5683 ↾ cres 5684 “ cima 5685 ∘ ccom 5686 ⟶wf 6549 –onto→wfo 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 |
This theorem is referenced by: fcoresfob 46483 funfocofob 46487 |
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