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Mirrors > Home > ILE Home > Th. List > ereldm | GIF version |
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ereldm.1 | ⊢ (𝜑 → 𝑅 Er 𝑋) |
ereldm.2 | ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) |
Ref | Expression |
---|---|
ereldm | ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ereldm.2 | . . . . 5 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
2 | 1 | eleq2d 2107 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅)) |
3 | 2 | exbidv 1706 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)) |
4 | ecdmn0m 6148 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅) | |
5 | ecdmn0m 6148 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅) | |
6 | 3, 4, 5 | 3bitr4g 212 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) |
7 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
8 | erdm 6116 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
10 | 9 | eleq2d 2107 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) |
11 | 9 | eleq2d 2107 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) |
12 | 6, 10, 11 | 3bitr3d 207 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 dom cdm 4345 Er wer 6103 [cec 6104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-er 6106 df-ec 6108 |
This theorem is referenced by: erth 6150 brecop 6196 |
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