Theorem disjlem19 38761

Description: Lemma for disjdmqseq 38765, partim2 38767 and petlem 38772 via disjdmqs 38764, (general version of the former prtlem19 38838). (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjlem19	⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑉

Proof of Theorem disjlem19

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjlem18 38760	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
2	1	elvd 3469	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
3	2	imp31 417	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
4		elecALTV 38226	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
5	4	elvd 3469	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
6	5	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
7	3, 6	bitr4d 282	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝑧 ∈ [𝐴] ≀ 𝑅))
8	7	eqrdv 2732	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)
9	8	exp31 419	1 ⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463   class class class wbr 5123  dom cdm 5665  [cec 8725   ≀ ccoss 38141   Disj wdisjALTV 38175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rmo 3363  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8729  df-coss 38371  df-cnvrefrel 38487  df-disjALTV 38665

This theorem is referenced by:  disjdmqsss  38762  disjdmqscossss  38763  eldisjlem19  38770