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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for fcores 44176. (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 6524 | . . . 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 4130 | . . . . 5 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ 𝐶 | |
6 | 4, 5 | eqsstrdi 3941 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ 𝐶) |
7 | 2, 6 | fnssresd 6479 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐸) Fn 𝐸) |
8 | fcores.y | . . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
9 | 8 | fneq1i 6454 | . . 3 ⊢ (𝑌 Fn 𝐸 ↔ (𝐺 ↾ 𝐸) Fn 𝐸) |
10 | 7, 9 | sylibr 237 | . 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 44174 | . . 3 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
15 | fofn 6613 | . . 3 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋 Fn 𝑃) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 Fn 𝑃) |
17 | 11, 3, 12, 13 | fcoreslem2 44173 | . . 3 ⊢ (𝜑 → ran 𝑋 = 𝐸) |
18 | eqimss 3943 | . . 3 ⊢ (ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑋 ⊆ 𝐸) |
20 | fnco 6472 | . 2 ⊢ ((𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸) → (𝑌 ∘ 𝑋) Fn 𝑃) | |
21 | 10, 16, 19, 20 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∩ cin 3852 ⊆ wss 3853 ◡ccnv 5535 ran crn 5537 ↾ cres 5538 “ cima 5539 ∘ ccom 5540 Fn wfn 6353 ⟶wf 6354 –onto→wfo 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 |
This theorem is referenced by: fcores 44176 |
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