Theorem ecuncnvepres 39327

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 39326	. . 3 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)))
2		elecreseq 8767	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅)
3		eccnvepres2 39223	. . . 4 ⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵)
4	2, 3	uneq12d 4116	. . 3 ⊢ (𝐵 ∈ 𝐴 → ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](◡ E ↾ 𝐴)) = ([𝐵]𝑅 ∪ 𝐵))
5	1, 4	eqtrd 2796	. 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = ([𝐵]𝑅 ∪ 𝐵))
6		uncom 4105	. 2 ⊢ ([𝐵]𝑅 ∪ 𝐵) = (𝐵 ∪ [𝐵]𝑅)
7	5, 6	eqtrdi 2812	1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∪ cun 3897   E cep 5550  ^◡ccnv 5650   ↾ cres 5653  [cec 8715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8719

This theorem is used by:  dfadjliftmap2  39389  blockadjliftmap  39390