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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmecd | Structured version Visualization version GIF version |
Description: Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8337). (Contributed by Peter Mazsa, 9-Oct-2018.) |
Ref | Expression |
---|---|
dmecd.1 | ⊢ (𝜑 → dom 𝑅 = 𝐴) |
dmecd.2 | ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅) |
Ref | Expression |
---|---|
dmecd | ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmecd.2 | . . . 4 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅) | |
2 | 1 | neeq1d 3075 | . . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅)) |
3 | ecdmn0 8336 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
4 | ecdmn0 8336 | . . 3 ⊢ (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 316 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅)) |
6 | dmecd.1 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) | |
7 | 6 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴)) |
8 | 6 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐶 ∈ dom 𝑅 ↔ 𝐶 ∈ 𝐴)) |
9 | 5, 7, 8 | 3bitr3d 311 | 1 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 dom cdm 5555 [cec 8287 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ec 8291 |
This theorem is referenced by: dmec2d 35578 |
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