Theorem disjlem19 38847

Description: Lemma for disjdmqseq 38851, partim2 38853 and petlem 38858 via disjdmqs 38850, (general version of the former prtlem19 38925). (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 38846	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
2	1	elvd 3442	. . . . 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 38309	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
5	4	elvd 3442	. . . . 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 2729	. 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 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  dom cdm 5614  [cec 8620   ≀ ccoss 38223   Disj wdisjALTV 38257

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

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-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  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-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-coss 38456  df-cnvrefrel 38572  df-disjALTV 38751

This theorem is referenced by:  disjdmqsss  38848  disjdmqscossss  38849  eldisjlem19  38856