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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 8320). (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 3046 | . . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅)) |
3 | ecdmn0 8319 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
4 | ecdmn0 8319 | . . 3 ⊢ (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 317 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅)) |
6 | dmecd.1 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴) | |
7 | 6 | eleq2d 2875 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴)) |
8 | 6 | eleq2d 2875 | . 2 ⊢ (𝜑 → (𝐶 ∈ dom 𝑅 ↔ 𝐶 ∈ 𝐴)) |
9 | 5, 7, 8 | 3bitr3d 312 | 1 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 dom cdm 5519 [cec 8270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 |
This theorem is referenced by: dmec2d 35723 |
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