Theorem ecuncnvepres 38894

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 38893	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)))
2		elecreseq 8728	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3		eccnvepres2 38790	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)
4	2, 3	uneq12d 4122	. . 3 ⊢ (𝐵 ∈ 𝐴 → ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)) = ([𝐵]𝑅 ∪ 𝐵))
5	1, 4	eqtrd 2797	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵]𝑅 ∪ 𝐵))
6		uncom 4111	. 2 ⊢ ([𝐵]𝑅 ∪ 𝐵) = (𝐵 ∪ [𝐵]𝑅)
7	5, 6	eqtrdi 2813	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   ∪ cun 3902   E cep 5546  ^◡ccnv 5646   ↾ cres 5649  [cec 8676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680

This theorem is referenced by:  dfadjliftmap2  38956  blockadjliftmap  38957