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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 6979 | . . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
4 | 3 | breqd 5160 | 1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1535 ∈ wcel 2104 Vcvv 3477 class class class wbr 5149 I cid 5575 ‘cfv 6558 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fv 6566 |
This theorem is referenced by: (None) |
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