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Mirrors > Home > MPE Home > Th. List > Mathboxes > eubrdm | Structured version Visualization version GIF version |
Description: If there is a unique set which is related to a class, then the class is an element of the domain of the relation. (Contributed by AV, 25-Aug-2022.) |
Ref | Expression |
---|---|
eubrdm | ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eubrv 46340 | . 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V) | |
2 | iotaex 6515 | . . 3 ⊢ (℩𝑏𝐴𝑅𝑏) ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V) |
4 | iota4 6523 | . . 3 ⊢ (∃!𝑏 𝐴𝑅𝑏 → [(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏) | |
5 | sbcbr12g 5198 | . . . . 5 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 ↔ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏)) | |
6 | 2, 5 | ax-mp 5 | . . . 4 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 ↔ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏) |
7 | csbconstg 3908 | . . . . . 6 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴) | |
8 | 2, 7 | ax-mp 5 | . . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴 |
9 | 2 | csbvargi 4428 | . . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 = (℩𝑏𝐴𝑅𝑏) |
10 | 8, 9 | breq12i 5151 | . . . 4 ⊢ (⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 ↔ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) |
11 | 6, 10 | sylbb 218 | . . 3 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) |
12 | 4, 11 | syl 17 | . 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) |
13 | breldmg 5906 | . 2 ⊢ ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅) | |
14 | 1, 3, 12, 13 | syl3anc 1369 | 1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∃!weu 2557 Vcvv 3469 [wsbc 3774 ⦋csb 3889 class class class wbr 5142 dom cdm 5672 ℩cio 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-dm 5682 df-iota 6494 |
This theorem is referenced by: dfafv2 46435 |
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