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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvilbd | Structured version Visualization version GIF version |
Description: A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) |
Ref | Expression |
---|---|
fvilbd.r | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
fvilbd | ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3948 | . 2 ⊢ 𝑅 ⊆ 𝑅 | |
2 | fvilbd.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
3 | fvi 6841 | . . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅) |
5 | 1, 4 | sseqtrrid 3979 | 1 ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 I cid 5489 ‘cfv 6432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-iota 6390 df-fun 6434 df-fv 6440 |
This theorem is referenced by: (None) |
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