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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 4176 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
2 | dfdm4 5757 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
4 | 3 | rneqd 5801 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
5 | rn0 5789 | . . . . . 6 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | syl6eq 2870 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
7 | 2, 6 | syl5eq 2866 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
8 | ineq2 4181 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
9 | in0 4343 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | syl6eq 2870 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
12 | 1, 11 | syl5eq 2866 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
13 | 12 | coemptyd 14331 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∩ cin 3933 ∅c0 4289 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 |
This theorem is referenced by: (None) |
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