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| 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 4774 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
| 3 | xpco 6309 | . . 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 5875 | . 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 2785 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉}) = {〈𝐶, 𝐵〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 〈cop 4632 × cxp 5683 ∘ ccom 5689 | 
| 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-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 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-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 | 
| This theorem is referenced by: coprprop 32708 | 
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