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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 4773 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | xpco 6281 | . . 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 7132 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}) | |
7 | 1, 5, 6 | syl2anc 583 | . . 3 ⊢ (𝜑 → ({𝐴} × {𝐵}) = {⟨𝐴, 𝐵⟩}) |
8 | cosnop.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
9 | xpsng 7132 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩}) | |
10 | 8, 1, 9 | syl2anc 583 | . . 3 ⊢ (𝜑 → ({𝐶} × {𝐴}) = {⟨𝐶, 𝐴⟩}) |
11 | 7, 10 | coeq12d 5857 | . 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩})) |
12 | xpsng 7132 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩}) | |
13 | 8, 5, 12 | syl2anc 583 | . 2 ⊢ (𝜑 → ({𝐶} × {𝐵}) = {⟨𝐶, 𝐵⟩}) |
14 | 4, 11, 13 | 3eqtr3d 2774 | 1 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 {csn 4623 ⟨cop 4629 × cxp 5667 ∘ ccom 5673 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 |
This theorem is referenced by: coprprop 32426 |
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