Theorem eccnvepres3 35544

Description: Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021.)

Assertion

Ref	Expression
eccnvepres3	⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Proof of Theorem eccnvepres3

Step	Hyp	Ref	Expression
1		resdmres 6091	. . 3 ⊢ (◡ E ↾ dom (◡ E ↾ 𝐴)) = (◡ E ↾ 𝐴)
2	1	eceq2i 8332	. 2 ⊢ [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = [𝐵](◡ E ↾ 𝐴)
3		eccnvepres2 35543	. 2 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ dom (◡ E ↾ 𝐴)) = 𝐵)
4	2, 3	syl5eqr 2872	1 ⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   E cep 5466  ^◡ccnv 5556  dom cdm 5557   ↾ cres 5559  [cec 8289

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-ne 3019  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-br 5069  df-opab 5131  df-eprel 5467  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ec 8293

This theorem is referenced by: (None)