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Mirrors > Home > MPE Home > Th. List > Mathboxes > conrel2d | 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 |
---|---|
conrel2d | ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5547 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | 1 | ineq2i 4110 | . . . 4 ⊢ (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom ◡𝐴) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom ◡𝐴)) |
4 | conrel1d.a | . . . . 5 ⊢ (𝜑 → ◡𝐴 = ∅) | |
5 | 4 | dmeqd 5759 | . . . 4 ⊢ (𝜑 → dom ◡𝐴 = dom ∅) |
6 | 5 | ineq2d 4113 | . . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ◡𝐴) = (dom 𝐵 ∩ dom ∅)) |
7 | dm0 5774 | . . . . . 6 ⊢ dom ∅ = ∅ | |
8 | 7 | ineq2i 4110 | . . . . 5 ⊢ (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅) |
9 | in0 4292 | . . . . 5 ⊢ (dom 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | eqtri 2759 | . . . 4 ⊢ (dom 𝐵 ∩ dom ∅) = ∅ |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅) |
12 | 3, 6, 11 | 3eqtrd 2775 | . 2 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅) |
13 | 12 | coemptyd 14507 | 1 ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∩ cin 3852 ∅c0 4223 ◡ccnv 5535 dom cdm 5536 ran crn 5537 ∘ ccom 5540 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 |
This theorem is referenced by: (None) |
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