Theorem disjlem18 37670

Description: Lemma for disjdmqseq 37675, partim2 37677 and petlem 37682 via disjlem19 37671, (general version of the former prtlem18 37747). (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjlem18	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝐵))))

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

Proof of Theorem disjlem18

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 3247	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)) → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))
2	1	expr 458	. . . . . 6 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
3	2	adantl 483	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝐵 ∈ [𝑥]𝑅 → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
4		relbrcoss 37316	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅))))
5		disjrel 37600	. . . . . . 7 ⊢ ( Disj 𝑅 → Rel 𝑅)
6	4, 5	impel 507	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
7	6	adantr 482	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))
8	3, 7	sylibrd 259	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (𝐵 ∈ [𝑥]𝑅 → 𝐴 ≀ 𝑅𝐵))
9	8	ex 414	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 → 𝐴 ≀ 𝑅𝐵)))
10		disjlem17 37669	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
11	10	adantl 483	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
12		relbrcoss 37316	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅))))
13	12, 5	impel 507	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)))
14	13	imbi1d 342	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → ((𝐴 ≀ 𝑅𝐵 → 𝐵 ∈ [𝑥]𝑅) ↔ (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
15	11, 14	sylibrd 259	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐴 ≀ 𝑅𝐵 → 𝐵 ∈ [𝑥]𝑅)))
16	9, 15	impbidd 209	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝐵)))
17	16	ex 414	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐴 ≀ 𝑅𝐵))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  ∃wrex 3071   class class class wbr 5149  dom cdm 5677  Rel wrel 5682  [cec 8701   ≀ ccoss 37043   Disj wdisjALTV 37077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705  df-coss 37281  df-cnvrefrel 37397  df-disjALTV 37575

This theorem is referenced by:  disjlem19  37671