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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fcores 47017. (Contributed by AV, 13-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) |
Ref | Expression |
---|---|
fcoreslem3 | ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcores.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6738 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | fcores.e | . . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐶)) |
5 | fcores.p | . . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐶)) |
7 | 2, 4, 6 | rescnvimafod 7093 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑃):𝑃–onto→𝐸) |
8 | fcores.x | . . 3 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
9 | foeq1 6817 | . . 3 ⊢ (𝑋 = (𝐹 ↾ 𝑃) → (𝑋:𝑃–onto→𝐸 ↔ (𝐹 ↾ 𝑃):𝑃–onto→𝐸)) | |
10 | 8, 9 | mp1i 13 | . 2 ⊢ (𝜑 → (𝑋:𝑃–onto→𝐸 ↔ (𝐹 ↾ 𝑃):𝑃–onto→𝐸)) |
11 | 7, 10 | mpbird 257 | 1 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
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 ⟶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-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-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 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-br 5149 df-opab 5211 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-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 |
This theorem is referenced by: fcoreslem4 47016 fcores 47017 fcoresf1lem 47018 fcoresf1 47019 fcoresfo 47021 fcoresfob 47022 |
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