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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 6165 | . . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
4 | 3 | breqd 4594 | 1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 I cid 4948 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: (None) |
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