dmxrnuncnvepres - Mathbox for Peter Mazsa

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Theorem dmxrnuncnvepres 38426

Description: Domain of the range Cartesian product with the converse epsilon relation combined with the union with the converse epsilon, restricted. (Contributed by Peter Mazsa, 28-Jan-2026.)

**Assertion**
Ref	Expression
dmxrnuncnvepres	⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})

**Proof of Theorem dmxrnuncnvepres**
Step	Hyp	Ref	Expression
1		dmuncnvepres 38425	. 2 ⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})))
2		dmxrncnvep 38423	. . . . 5 ⊢ dom (𝑅 ⋉ ^◡ E ) = (dom 𝑅 ∖ {∅})
3	2	uneq1i 4111	. . . 4 ⊢ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
4		difundir 4238	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
5		unv 4346	. . . . 5 ⊢ (dom 𝑅 ∪ V) = V
6	5	difeq1i 4069	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = (V ∖ {∅})
7	3, 4, 6	3eqtr2i 2760	. . 3 ⊢ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})) = (V ∖ {∅})
8	7	ineq2i 4164	. 2 ⊢ (𝐴 ∩ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅}))) = (𝐴 ∩ (V ∖ {∅}))
9		invdif 4226	. 2 ⊢ (𝐴 ∩ (V ∖ {∅})) = (𝐴 ∖ {∅})
10	1, 8, 9	3eqtri 2758	1 ⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 Vcvv 3436 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ∅c0 4280 {csn 4573 E cep 5513 ^◡ccnv 5613 dom cdm 5614 ↾ cres 5616 ⋉ cxrn 38224

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

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-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-eprel 5514 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-oprab 7350 df-1st 7921 df-2nd 7922 df-xrn 38414

This theorem is referenced by: blockadjliftmap 38482

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