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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 6076 | . . . . 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 6084 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → ran {〈𝐶, 𝐷〉} = {𝐷}) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → ran {〈𝐶, 𝐷〉} = {𝐷}) |
7 | 3, 6 | ineq12d 4128 | . . 3 ⊢ (𝜑 → (dom {〈𝐴, 𝐵〉} ∩ ran {〈𝐶, 𝐷〉}) = ({𝐴} ∩ {𝐷})) |
8 | cosnopne.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐷) | |
9 | disjsn2 4628 | . . . 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 14542 | 1 ⊢ (𝜑 → ({〈𝐴, 𝐵〉} ∘ {〈𝐶, 𝐷〉}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∩ cin 3865 ∅c0 4237 {csn 4541 〈cop 4547 dom cdm 5551 ran crn 5552 ∘ ccom 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 |
This theorem is referenced by: coprprop 30752 |
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