Theorem fvilbd 44688

Description: A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)

Hypothesis

Ref	Expression
fvilbd.r	⊢ (𝜑 → 𝑅 ∈ V)

Assertion

Ref	Expression
fvilbd	⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))

Proof of Theorem fvilbd

Step	Hyp	Ref	Expression
1		ssid 3953	. 2 ⊢ 𝑅 ⊆ 𝑅
2		fvilbd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		fvi 6961	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
4	2, 3	syl 18	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
5	1, 4	sseqtrrid 3974	1 ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899   I cid 5545  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by: (None)