dmxrnuncnvepres - Mathbox for Peter Mazsa

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Theorem dmxrnuncnvepres 38516

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 38515	. 2 ⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})))
2		dmxrncnvep 38513	. . . . 5 ⊢ dom (𝑅 ⋉ ^◡ E ) = (dom 𝑅 ∖ {∅})
3	2	uneq1i 4114	. . . 4 ⊢ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
4		difundir 4241	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
5		unv 4349	. . . . 5 ⊢ (dom 𝑅 ∪ V) = V
6	5	difeq1i 4072	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = (V ∖ {∅})
7	3, 4, 6	3eqtr2i 2763	. . 3 ⊢ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅})) = (V ∖ {∅})
8	7	ineq2i 4167	. 2 ⊢ (𝐴 ∩ (dom (𝑅 ⋉ ^◡ E ) ∪ (V ∖ {∅}))) = (𝐴 ∩ (V ∖ {∅}))
9		invdif 4229	. 2 ⊢ (𝐴 ∩ (V ∖ {∅})) = (𝐴 ∖ {∅})
10	1, 8, 9	3eqtri 2761	1 ⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 Vcvv 3438 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ∅c0 4283 {csn 4578 E cep 5521 ^◡ccnv 5621 dom cdm 5622 ↾ cres 5624 ⋉ cxrn 38314

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-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-un 7678

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-eprel 5522 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-fo 6496 df-fv 6498 df-oprab 7360 df-1st 7931 df-2nd 7932 df-xrn 38504

This theorem is referenced by: blockadjliftmap 38572

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