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Mirrors > Home > MPE Home > Th. List > ereldm | Structured version Visualization version 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 | . . . 4 ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅) | |
2 | 1 | neeq1d 2853 | . . 3 ⊢ (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅)) |
3 | ecdmn0 7789 | . . 3 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅) | |
4 | ecdmn0 7789 | . . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅) | |
5 | 2, 3, 4 | 3bitr4g 303 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅)) |
6 | ereldm.1 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
7 | erdm 7752 | . . . 4 ⊢ (𝑅 Er 𝑋 → dom 𝑅 = 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝑋) |
9 | 8 | eleq2d 2687 | . 2 ⊢ (𝜑 → (𝐴 ∈ dom 𝑅 ↔ 𝐴 ∈ 𝑋)) |
10 | 8 | eleq2d 2687 | . 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝑋)) |
11 | 5, 9, 10 | 3bitr3d 298 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 dom cdm 5114 Er wer 7739 [cec 7740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-er 7742 df-ec 7744 |
This theorem is referenced by: erth 7791 brecop 7840 eceqoveq 7853 |
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