Theorem fvilbd 38821

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 3847	. 2 ⊢ 𝑅 ⊆ 𝑅
2		fvilbd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		fvi 6501	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
5	1, 4	syl5sseqr 3878	1 ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3413   ⊆ wss 3797   I cid 5248  ‘cfv 6122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-iota 6085  df-fun 6124  df-fv 6130

This theorem is referenced by: (None)