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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoreslem2 | Structured version Visualization version GIF version |
Description: Lemma 2 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) |
Ref | Expression |
---|---|
fcoreslem2 | ⊢ (𝜑 → ran 𝑋 = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5549 | . . 3 ⊢ (𝐹 “ 𝑃) = ran (𝐹 ↾ 𝑃) | |
2 | fcores.x | . . . . . 6 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
3 | 2 | rneqi 5791 | . . . . 5 ⊢ ran 𝑋 = ran (𝐹 ↾ 𝑃) |
4 | 3 | eqcomi 2745 | . . . 4 ⊢ ran (𝐹 ↾ 𝑃) = ran 𝑋 |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝑃) = ran 𝑋) |
6 | 1, 5 | eqtr2id 2784 | . 2 ⊢ (𝜑 → ran 𝑋 = (𝐹 “ 𝑃)) |
7 | fcores.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
8 | fcores.e | . . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
9 | fcores.p | . . . 4 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
10 | 7, 8, 9 | fcoreslem1 44172 | . . 3 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ 𝐸)) |
11 | 10 | imaeq2d 5914 | . 2 ⊢ (𝜑 → (𝐹 “ 𝑃) = (𝐹 “ (◡𝐹 “ 𝐸))) |
12 | 7 | ffund 6527 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
13 | funimacnv 6439 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = (𝐸 ∩ ran 𝐹)) |
15 | inss1 4129 | . . . . . 6 ⊢ (ran 𝐹 ∩ 𝐶) ⊆ ran 𝐹 | |
16 | 8, 15 | eqsstri 3921 | . . . . 5 ⊢ 𝐸 ⊆ ran 𝐹 |
17 | 16 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐸 ⊆ ran 𝐹) |
18 | df-ss 3870 | . . . 4 ⊢ (𝐸 ⊆ ran 𝐹 ↔ (𝐸 ∩ ran 𝐹) = 𝐸) | |
19 | 17, 18 | sylib 221 | . . 3 ⊢ (𝜑 → (𝐸 ∩ ran 𝐹) = 𝐸) |
20 | 14, 19 | eqtrd 2771 | . 2 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐸)) = 𝐸) |
21 | 6, 11, 20 | 3eqtrd 2775 | 1 ⊢ (𝜑 → ran 𝑋 = 𝐸) |
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 Fun wfun 6352 ⟶wf 6354 |
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 |
This theorem is referenced by: fcoreslem4 44175 fcoresf1 44178 |
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