Theorem disjlem17 38755

Description: Lemma for disjdmqseq 38761, partim2 38763 and petlem 38768 via disjlem18 38756, (general version of the former prtlem17 38832). (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 3077	. . 3 ⊢ (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)))
2		an32 645	. . . . . . . 8 ⊢ (((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3		disjlem14 38754	. . . . . . . . . . 11 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4		eleq2 2833	. . . . . . . . . . . 12 ⊢ ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐵 ∈ [𝑦]𝑅))
5	4	biimprd 248	. . . . . . . . . . 11 ⊢ ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))
6	3, 5	syl8 76	. . . . . . . . . 10 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
7	6	exp4a 431	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
8	7	impd 410	. . . . . . . 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 243	. . . . . . 7 ⊢ ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
10	9	expd 415	. . . . . 6 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
11	10	imp5a 440	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
12	11	imp4b 421	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
13	12	exlimdv 1932	. . 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 242	. 2 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
15	14	ex 412	1 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  dom cdm 5700  [cec 8761   Disj wdisjALTV 38169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-coss 38367  df-cnvrefrel 38483  df-disjALTV 38661

This theorem is referenced by:  disjlem18  38756