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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cofid1 | Structured version Visualization version GIF version | ||
| Description: Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| Ref | Expression |
|---|---|
| cofid1a.i | ⊢ 𝐼 = (idfunc‘𝐷) |
| cofid1a.b | ⊢ 𝐵 = (Base‘𝐷) |
| cofid1a.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| cofid1.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
| cofid1.k | ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿) |
| cofid1.o | ⊢ (𝜑 → (〈𝐾, 𝐿〉 ∘func 〈𝐹, 𝐺〉) = 𝐼) |
| Ref | Expression |
|---|---|
| cofid1 | ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cofid1.k | . . . 4 ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿) | |
| 2 | 1 | func1st 49707 | . . 3 ⊢ (𝜑 → (1st ‘〈𝐾, 𝐿〉) = 𝐾) |
| 3 | cofid1.f | . . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
| 4 | 3 | func1st 49707 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝐹, 𝐺〉) = 𝐹) |
| 5 | 4 | fveq1d 6873 | . . 3 ⊢ (𝜑 → ((1st ‘〈𝐹, 𝐺〉)‘𝑋) = (𝐹‘𝑋)) |
| 6 | 2, 5 | fveq12d 6878 | . 2 ⊢ (𝜑 → ((1st ‘〈𝐾, 𝐿〉)‘((1st ‘〈𝐹, 𝐺〉)‘𝑋)) = (𝐾‘(𝐹‘𝑋))) |
| 7 | cofid1a.i | . . 3 ⊢ 𝐼 = (idfunc‘𝐷) | |
| 8 | cofid1a.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
| 9 | cofid1a.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 10 | df-br 5105 | . . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
| 11 | 3, 10 | sylib 221 | . . 3 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
| 12 | df-br 5105 | . . . 4 ⊢ (𝐾(𝐸 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐸 Func 𝐷)) | |
| 13 | 1, 12 | sylib 221 | . . 3 ⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐸 Func 𝐷)) |
| 14 | cofid1.o | . . 3 ⊢ (𝜑 → (〈𝐾, 𝐿〉 ∘func 〈𝐹, 𝐺〉) = 𝐼) | |
| 15 | 7, 8, 9, 11, 13, 14 | cofid1a 49742 | . 2 ⊢ (𝜑 → ((1st ‘〈𝐾, 𝐿〉)‘((1st ‘〈𝐹, 𝐺〉)‘𝑋)) = 𝑋) |
| 16 | 6, 15 | eqtr3d 2802 | 1 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1563 ∈ wcel 2145 〈cop 4591 class class class wbr 5104 ‘cfv 6525 (class class class)co 7400 1st c1st 7972 Basecbs 17257 Func cfunc 17899 idfunccidfu 17900 ∘func ccofu 17901 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975 df-map 8814 df-ixp 8884 df-func 17903 df-idfu 17904 df-cofu 17905 |
| This theorem is referenced by: (None) |
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