Theorem disjlem17 39651

Description: Lemma for disjdmqseq 39657, partim2 39659 and petlem 39664 via disjlem18 39652, (general version of the former prtlem17 39750). (Contributed by Peter Mazsa, 10-Sep-2021.)

Assertion

Ref	Expression
disjlem17	⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjlem17

Step	Hyp	Ref	Expression
1		df-rex 3087	. . 3 ⊢ (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)))
2		an32 659	. . . . . . . 8 ⊢ (((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3		disjlem14 39650	. . . . . . . . . . 11 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4		eleq2 2849	. . . . . . . . . . . 12 ⊢ ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐵 ∈ [𝑦]𝑅))
5	4	biimprd 251	. . . . . . . . . . 11 ⊢ ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))
6	3, 5	syl8 77	. . . . . . . . . 10 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
7	6	exp4a 437	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
8	7	impd 416	. . . . . . . 8 ⊢ ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
9	2, 8	biimtrrid 246	. . . . . . 7 ⊢ ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
10	9	expd 421	. . . . . 6 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
11	10	imp5a 446	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
12	11	imp4b 427	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
13	12	exlimdv 1966	. . 3 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
14	1, 13	biimtrid 245	. 2 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
15	14	ex 418	1 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3086  dom cdm 5655  [cec 8695   Disj wdisjALTV 38968

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 2732  ax-sep 5251  ax-pr 5398

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 2564  df-eu 2594  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-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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8699  df-coss 39250  df-cnvrefrel 39356  df-disjALTV 39539

This theorem is used by:  disjlem18  39652