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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 4092 | . . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴) | |
2 | dfdm4 5739 | . . . . 5 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | conrel1d.a | . . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅) | |
4 | 3 | rneqd 5782 | . . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅) |
5 | rn0 5770 | . . . . . 6 ⊢ ran ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2790 | . . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅) |
7 | 2, 6 | syl5eq 2786 | . . . 4 ⊢ (𝜑 → dom 𝐴 = ∅) |
8 | ineq2 4098 | . . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅)) | |
9 | in0 4281 | . . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅ | |
10 | 8, 9 | eqtrdi 2790 | . . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅) |
12 | 1, 11 | syl5eq 2786 | . 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) |
13 | 12 | coemptyd 14431 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∩ cin 3843 ∅c0 4212 ◡ccnv 5525 dom cdm 5526 ran crn 5527 ∘ ccom 5530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-br 5032 df-opab 5094 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 |
This theorem is referenced by: (None) |
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