Theorem disjlem19 39836

Description: Lemma for disjdmqseq 39840, partim2 39842 and petlem 39847 via disjdmqs 39839, (general version of the former prtlem19 39935). (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 39835	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝑧))))
2	1	elvd 3457	. . . . 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 39203	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝑧))
5	4	elvd 3457	. . . . 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 2759	. 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 1570   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  dom cdm 5651  [cec 8715   ≀ ccoss 39115   Disj wdisjALTV 39151

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  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 8719  df-coss 39433  df-cnvrefrel 39539  df-disjALTV 39722

This theorem is used by:  disjdmqsss  39837  disjdmqscossss  39838  eldisjlem19  39845