Theorem disjlem19 39973

Description: Lemma for disjdmqseq 39977, partim2 39979 and petlem 39984 via disjdmqs 39976, (general version of the former prtlem19 40072). (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 39972	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
2	1	elvd 3456	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
3	2	imp31 423	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
4		elecALTV 39340	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
5	4	elvd 3456	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
6	5	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
7	3, 6	bitr4d 285	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝑧 ∈ [𝐴] ≀ 𝑅))
8	7	eqrdv 2758	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)
9	8	exp31 425	1 ⊢ (𝐴 ∈ 𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102  dom cdm 5651  [cec 8723   ≀ ccoss 39252   Disj wdisjALTV 39288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8727  df-coss 39570  df-cnvrefrel 39676  df-disjALTV 39859

This theorem is used by:  disjdmqsss  39974  disjdmqscossss  39975  eldisjlem19  39982