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 4710 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | xpco 6140 | . . 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 6901 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
7 | 1, 5, 6 | syl2anc 586 | . . 3 ⊢ (𝜑 → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) |
8 | cosnop.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
9 | xpsng 6901 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → ({𝐶} × {𝐴}) = {〈𝐶, 𝐴〉}) | |
10 | 8, 1, 9 | syl2anc 586 | . . 3 ⊢ (𝜑 → ({𝐶} × {𝐴}) = {〈𝐶, 𝐴〉}) |
11 | 7, 10 | coeq12d 5735 | . 2 ⊢ (𝜑 → (({𝐴} × {𝐵}) ∘ ({𝐶} × {𝐴})) = ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉})) |
12 | xpsng 6901 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑊) → ({𝐶} × {𝐵}) = {〈𝐶, 𝐵〉}) | |
13 | 8, 5, 12 | syl2anc 586 | . 2 ⊢ (𝜑 → ({𝐶} × {𝐵}) = {〈𝐶, 𝐵〉}) |
14 | 4, 11, 13 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐴〉}) = {〈𝐶, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 {csn 4567 〈cop 4573 × cxp 5553 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 |
This theorem is referenced by: coprprop 30435 |
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