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Mathbox for Thierry Arnoux |
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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 6235 | . . . . 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 6243 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {〈𝐶, 𝐷〉} = {𝐷}) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → ran {〈𝐶, 𝐷〉} = {𝐷}) |
7 | 3, 6 | ineq12d 4229 | . . 3 ⊢ (𝜑 → (dom {〈𝐴, 𝐵〉} ∩ ran {〈𝐶, 𝐷〉}) = ({𝐴} ∩ {𝐷})) |
8 | cosnopne.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷) | |
9 | disjsn2 4717 | . . . 4 ⊢ (𝐴 ≠ 𝐷 → ({𝐴} ∩ {𝐷}) = ∅) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐷}) = ∅) |
11 | 7, 10 | eqtrd 2775 | . 2 ⊢ (𝜑 → (dom {〈𝐴, 𝐵〉} ∩ ran {〈𝐶, 𝐷〉}) = ∅) |
12 | 11 | coemptyd 15015 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐷〉}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∩ cin 3962 ∅c0 4339 {csn 4631 〈cop 4637 dom cdm 5689 ran crn 5690 ∘ 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-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 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-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 |
This theorem is referenced by: coprprop 32714 |
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