| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cofmpt2 | Structured version Visualization version GIF version | ||
| Description: Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.) |
| Ref | Expression |
|---|---|
| cofmpt2.1 | ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) |
| cofmpt2.2 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸) |
| cofmpt2.3 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| cofmpt2.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| cofmpt2 | ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cofmpt2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸) | |
| 2 | 1 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸) |
| 3 | cofmpt2.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 4 | fcompt 7153 | . . 3 ⊢ (((𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸 ∧ 𝐹:𝐴⟶𝐵) → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)))) | |
| 5 | 2, 3, 4 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)))) |
| 6 | eqid 2737 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 7 | cofmpt2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) | |
| 8 | 7 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) |
| 9 | 3 | ffvelcdmda 7104 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
| 10 | cofmpt2.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
| 11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝑉) |
| 12 | 6, 8, 9, 11 | fvmptd2 7024 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)) = 𝐷) |
| 13 | 12 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
| 14 | 5, 13 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |