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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for fcores 46917. (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) |
Ref | Expression |
---|---|
fcoreslem4 | ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcores.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
2 | 1 | ffnd 6747 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐶) |
3 | fcores.e | . . . . . 6 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐶)) |
5 | inss2 4253 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
6 | 4, 5 | eqsstrdi 4057 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) |
7 | 2, 6 | fnssresd 6703 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸) |
8 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
9 | 8 | fneq1i 6675 | . . 3 ⊢ (𝑌 Fn 𝐸 ↔ (𝐺 ↾ 𝐸) Fn 𝐸) |
10 | 7, 9 | sylibr 234 | . 2 ⊢ (𝜑 → 𝑌 Fn 𝐸) |
11 | fcores.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
12 | fcores.p | . . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
13 | fcores.x | . . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
14 | 11, 3, 12, 13 | fcoreslem3 46915 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
15 | fofn 6835 | . . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 Fn 𝑃) |
17 | 11, 3, 12, 13 | fcoreslem2 46914 | . . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸) |
18 | eqimss 4061 | . . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸) |
20 | fnco 6696 | . 2 ⊢ ((𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸) → (𝑌 ∘ 𝑋) Fn 𝑃) | |
21 | 10, 16, 19, 20 | syl3anc 1371 | 1 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3969 ⊆ wss 3970 ◡ccnv 5698 ran crn 5700 ↾ cres 5701 “ cima 5702 ∘ ccom 5703 Fn wfn 6567 ⟶wf 6568 –onto→wfo 6570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-fun 6574 df-fn 6575 df-f 6576 df-fo 6578 |
This theorem is referenced by: fcores 46917 |
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