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Mathbox for Thierry Arnoux |
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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 7130 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐸) |
3 | cofmpt2.3 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
4 | fcompt 7148 | . . 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 582 | . 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)))) |
6 | eqid 2728 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶) | |
7 | cofmpt2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) | |
8 | 7 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷) |
9 | 3 | ffvelcdmda 7099 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
10 | cofmpt2.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
11 | 10 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝑉) |
12 | 6, 8, 9, 11 | fvmptd2 7018 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥)) = 𝐷) |
13 | 12 | mpteq2dva 5252 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘(𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
14 | 5, 13 | eqtrd 2768 | 1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5235 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 |
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-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 |
This theorem is referenced by: (None) |
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