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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosnop | Structured version Visualization version GIF version |
Description: Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
Ref | Expression |
---|---|
cosnop.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
cosnop.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
cosnop.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
Ref | Expression |
---|---|
cosnop | ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉}) = {〈𝐶, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosnop.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | snnzg 4779 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | xpco 6311 | . . 3 ⊢ ({𝐴} ≠ ∅ → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵})) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({𝐶} × {𝐵})) |
5 | cosnop.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
6 | xpsng 7159 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
7 | 1, 5, 6 | syl2anc 584 | . . 3 ⊢ (𝜑 → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
8 | cosnop.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
9 | xpsng 7159 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → ({𝐶} × {𝐴}) = {〈𝐶, 𝐴〉}) | |
10 | 8, 1, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → ({𝐶} × {𝐴}) = {〈𝐶, 𝐴〉}) |
11 | 7, 10 | coeq12d 5878 | . 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉})) |
12 | xpsng 7159 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ({𝐶} × {𝐵}) = {〈𝐶, 𝐵〉}) | |
13 | 8, 5, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → ({𝐶} × {𝐵}) = {〈𝐶, 𝐵〉}) |
14 | 4, 11, 13 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉}) = {〈𝐶, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {csn 4631 〈cop 4637 × cxp 5687 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: coprprop 32714 |
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