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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 5631 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | 1 | ineq2i 4156 | . . . 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 5847 | . . . 4 ⊢ (𝜑 → dom ◡𝐴 = dom ∅) |
6 | 5 | ineq2d 4159 | . . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ◡𝐴) = (dom 𝐵 ∩ dom ∅)) |
7 | dm0 5862 | . . . . . 6 ⊢ dom ∅ = ∅ | |
8 | 7 | ineq2i 4156 | . . . . 5 ⊢ (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅) |
9 | in0 4338 | . . . . 5 ⊢ (dom 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | eqtri 2764 | . . . 4 ⊢ (dom 𝐵 ∩ dom ∅) = ∅ |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅) |
12 | 3, 6, 11 | 3eqtrd 2780 | . 2 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅) |
13 | 12 | coemptyd 14789 | 1 ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∩ cin 3897 ∅c0 4269 ◡ccnv 5619 dom cdm 5620 ran crn 5621 ∘ ccom 5624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-br 5093 df-opab 5155 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 |
This theorem is referenced by: (None) |
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