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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for fcores 46984. (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 6750 | . . . 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 4259 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
6 | 4, 5 | eqsstrdi 4063 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) |
7 | 2, 6 | fnssresd 6706 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸) |
8 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
9 | 8 | fneq1i 6678 | . . 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 46982 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
15 | fofn 6838 | . . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 Fn 𝑃) |
17 | 11, 3, 12, 13 | fcoreslem2 46981 | . . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸) |
18 | eqimss 4067 | . . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸) |
20 | fnco 6699 | . 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 3975 ⊆ wss 3976 ◡ccnv 5699 ran crn 5701 ↾ cres 5702 “ cima 5703 ∘ ccom 5704 Fn wfn 6570 ⟶wf 6571 –onto→wfo 6573 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-fun 6577 df-fn 6578 df-f 6579 df-fo 6581 |
This theorem is referenced by: fcores 46984 |
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