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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 4135 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
2 | dfdm4 5804 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
4 | 3 | rneqd 5847 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
5 | rn0 5835 | . . . . . 6 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2794 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
7 | 2, 6 | eqtrid 2790 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
8 | ineq2 4140 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
9 | in0 4325 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | eqtrdi 2794 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
12 | 1, 11 | eqtrid 2790 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
13 | 12 | coemptyd 14690 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∩ cin 3886 ∅c0 4256 ◡ccnv 5588 dom cdm 5589 ran crn 5590 ∘ ccom 5593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 |
This theorem is referenced by: (None) |
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