Theorem dmxrnuncnvepres 39041

Description: Domain of the combined relation of two special relations, see blockadjliftmap 39107. (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 39040	. 2 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom (𝑅 ⋉ ◡ E ) ∪ (V ∖ {∅})))
2		dmxrncnvep 39038	. . . . 5 ⊢ dom (𝑅 ⋉ ◡ E ) = (dom 𝑅 ∖ {∅})
3	2	uneq1i 4118	. . . 4 ⊢ (dom (𝑅 ⋉ ◡ E ) ∪ (V ∖ {∅})) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
4		difundir 4244	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = ((dom 𝑅 ∖ {∅}) ∪ (V ∖ {∅}))
5		unv 4356	. . . . 5 ⊢ (dom 𝑅 ∪ V) = V
6	5	difeq1i 4077	. . . 4 ⊢ ((dom 𝑅 ∪ V) ∖ {∅}) = (V ∖ {∅})
7	3, 4, 6	3eqtr2i 2792	. . 3 ⊢ (dom (𝑅 ⋉ ◡ E ) ∪ (V ∖ {∅})) = (V ∖ {∅})
8	7	ineq2i 4170	. 2 ⊢ (𝐴 ∩ (dom (𝑅 ⋉ ◡ E ) ∪ (V ∖ {∅}))) = (𝐴 ∩ (V ∖ {∅}))
9		invdif 4232	. 2 ⊢ (𝐴 ∩ (V ∖ {∅})) = (𝐴 ∖ {∅})
10	1, 8, 9	3eqtri 2790	1 ⊢ dom (((𝑅 ⋉ ◡ E ) ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904  ∅c0 4286  {csn 4589   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663   ⋉ cxrn 38823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-oprab 7414  df-1st 7982  df-2nd 7983  df-xrn 39029

This theorem is referenced by:  blockadjliftmap  39107