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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for fcores 46512. (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) |
Ref | Expression |
---|---|
fcoreslem1 | ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcores.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffund 6721 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | cnvimainrn 7071 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐶)) = (◡𝐹 “ 𝐶)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (ran 𝐹 ∩ 𝐶)) = (◡𝐹 “ 𝐶)) |
5 | 4 | eqcomd 2731 | . 2 ⊢ (𝜑 → (◡𝐹 “ 𝐶) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶))) |
6 | fcores.p | . 2 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
7 | fcores.e | . . 3 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
8 | 7 | imaeq2i 6056 | . 2 ⊢ (◡𝐹 “ 𝐸) = (◡𝐹 “ (ran 𝐹 ∩ 𝐶)) |
9 | 5, 6, 8 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3938 ◡ccnv 5671 ran crn 5673 “ cima 5675 Fun wfun 6537 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: fcoreslem2 46509 |
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