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Mirrors > Home > MPE Home > Th. List > Mathboxes > brfvid | Structured version Visualization version GIF version |
Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.) |
Ref | Expression |
---|---|
brfvid.r | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
brfvid | ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brfvid.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | fvi 6983 | . . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
4 | 3 | breqd 5152 | 1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3479 class class class wbr 5141 I cid 5575 ‘cfv 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6512 df-fun 6561 df-fv 6567 |
This theorem is referenced by: (None) |
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