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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 Richard Penner, 24-Dec-2019.) |
Ref | Expression |
---|---|
conrel1d.a | ⊢ (𝜑 → ◡𝐴 = ∅) |
Ref | Expression |
---|---|
conrel1d | ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3767 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
2 | dfdm4 5238 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
4 | 3 | rneqd 5274 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
5 | rn0 5298 | . . . . . 6 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | syl6eq 2660 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
7 | 2, 6 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
8 | ineq2 3770 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
9 | in0 3920 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | syl6eq 2660 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
12 | 1, 11 | syl5eq 2656 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
13 | 12 | coemptyd 13566 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ∅c0 3874 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 |
This theorem is referenced by: (None) |
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