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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cosnopne | Structured version Visualization version GIF version | ||
| Description: Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| cosnopne.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| cosnopne.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) | 
| cosnopne.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐷) | 
| Ref | Expression | 
|---|---|
| cosnopne | ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐷〉}) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cosnopne.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 2 | dmsnopg 6233 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → dom {〈𝐴, 𝐵〉} = {𝐴}) | 
| 4 | cosnopne.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
| 5 | rnsnopg 6241 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {〈𝐶, 𝐷〉} = {𝐷}) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → ran {〈𝐶, 𝐷〉} = {𝐷}) | 
| 7 | 3, 6 | ineq12d 4221 | . . 3 ⊢ (𝜑 → (dom {〈𝐴, 𝐵〉} ∩ ran {〈𝐶, 𝐷〉}) = ({𝐴} ∩ {𝐷})) | 
| 8 | cosnopne.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷) | |
| 9 | disjsn2 4712 | . . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅) | |
| 10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅) | 
| 11 | 7, 10 | eqtrd 2777 | . 2 ⊢ (𝜑 → (dom {〈𝐴, 𝐵〉} ∩ ran {〈𝐶, 𝐷〉}) = ∅) | 
| 12 | 11 | coemptyd 15018 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐷〉}) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {csn 4626 〈cop 4632 dom cdm 5685 ran crn 5686 ∘ 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-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 | 
| This theorem is referenced by: coprprop 32708 | 
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