Theorem fvilbd 40041

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 3991	. 2 ⊢ 𝑅 ⊆ 𝑅
2		fvilbd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
3		fvi 6742	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
5	1, 4	sseqtrrid 4022	1 ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938   I cid 5461  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by: (None)