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Mirrors > Home > MPE Home > Th. List > Mathboxes > conrel1d | Structured version Visualization version GIF version |
Description: Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
conrel1d.a | ⊢ (𝜑 → ◡𝐴 = ∅) |
Ref | Expression |
---|---|
conrel1d | ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4203 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
2 | dfdm4 5902 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
4 | 3 | rneqd 5944 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
5 | rn0 5932 | . . . . . 6 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2784 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
7 | 2, 6 | eqtrid 2780 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
8 | ineq2 4208 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
9 | in0 4395 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | eqtrdi 2784 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
12 | 1, 11 | eqtrid 2780 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
13 | 12 | coemptyd 14966 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∩ cin 3948 ∅c0 4326 ◡ccnv 5681 dom cdm 5682 ran crn 5683 ∘ ccom 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 |
This theorem is referenced by: (None) |
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