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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for fcores 44539. (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 6593 | . . 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 6943 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑃):𝑃–onto→𝐸) |
8 | fcores.x | . . 3 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
9 | foeq1 6676 | . . 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 256 | 1 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∩ cin 3885 ◡ccnv 5583 ran crn 5585 ↾ cres 5586 “ cima 5587 ⟶wf 6422 –onto→wfo 6424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-fun 6428 df-fn 6429 df-f 6430 df-fo 6432 |
This theorem is referenced by: fcoreslem4 44538 fcores 44539 fcoresf1lem 44540 fcoresf1 44541 fcoresfo 44543 fcoresfob 44544 |
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