Theorem ecuncnvepres 38580

Description: The restricted union with converse epsilon relation coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
ecuncnvepres	⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))

Proof of Theorem ecuncnvepres

Step	Hyp	Ref	Expression
1		ecunres 38579	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)))
2		elecreseq 8684	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3		eccnvepres2 38484	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)
4	2, 3	uneq12d 4121	. . 3 ⊢ (𝐵 ∈ 𝐴 → ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)) = ([𝐵]𝑅 ∪ 𝐵))
5	1, 4	eqtrd 2771	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵]𝑅 ∪ 𝐵))
6		uncom 4110	. 2 ⊢ ([𝐵]𝑅 ∪ 𝐵) = (𝐵 ∪ [𝐵]𝑅)
7	5, 6	eqtrdi 2787	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   E cep 5523  ^◡ccnv 5623   ↾ cres 5626  [cec 8633

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  dfadjliftmap2  38632  blockadjliftmap  38633