Theorem fvilbd 44435

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 3959	. 2 ⊢ 𝑅 ⊆ 𝑅
2		fvilbd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		fvi 6957	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
4	2, 3	syl 18	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
5	1, 4	sseqtrrid 3980	1 ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905   I cid 5555  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is referenced by: (None)