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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 8748). (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 2992 | . . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅)) |
3 | ecdmn0 8747 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
4 | ecdmn0 8747 | . . 3 ⊢ (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅)) |
6 | dmecd.1 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) | |
7 | 6 | eleq2d 2811 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴)) |
8 | 6 | eleq2d 2811 | . 2 ⊢ (𝜑 → (𝐶 ∈ dom 𝑅 ↔ 𝐶 ∈ 𝐴)) |
9 | 5, 7, 8 | 3bitr3d 309 | 1 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∅c0 4315 dom cdm 5667 [cec 8698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 |
This theorem is referenced by: dmec2d 37668 |
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