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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcoresf1lem | Structured version Visualization version GIF version |
Description: Lemma for fcoresf1 44178. (Contributed by AV, 18-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 |
---|---|
fcoresf1lem | ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcores.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fcores.e | . . . . 5 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶) | |
3 | fcores.p | . . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶) | |
4 | fcores.x | . . . . 5 ⊢ 𝑋 = (𝐹 ↾ 𝑃) | |
5 | fcores.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷) | |
6 | fcores.y | . . . . 5 ⊢ 𝑌 = (𝐺 ↾ 𝐸) | |
7 | 1, 2, 3, 4, 5, 6 | fcores 44176 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋)) |
8 | 7 | fveq1d 6697 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍)) |
9 | 8 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = ((𝑌 ∘ 𝑋)‘𝑍)) |
10 | 1, 2, 3, 4 | fcoreslem3 44174 | . . . . 5 ⊢ (𝜑 → 𝑋:𝑃–onto→𝐸) |
11 | fof 6611 | . . . . 5 ⊢ (𝑋:𝑃–onto→𝐸 → 𝑋:𝑃⟶𝐸) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋:𝑃⟶𝐸) |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑋:𝑃⟶𝐸) |
14 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → 𝑍 ∈ 𝑃) | |
15 | 13, 14 | fvco3d 6789 | . 2 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝑌 ∘ 𝑋)‘𝑍) = (𝑌‘(𝑋‘𝑍))) |
16 | 9, 15 | eqtrd 2771 | 1 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ◡ccnv 5535 ran crn 5537 ↾ cres 5538 “ cima 5539 ∘ ccom 5540 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 |
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-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 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-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 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-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 |
This theorem is referenced by: fcoresf1 44178 |
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