Theorem disjlem17 38845

Description: Lemma for disjdmqseq 38851, partim2 38853 and petlem 38858 via disjlem18 38846, (general version of the former prtlem17 38923). (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 3057	. . 3 ⊢ (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)))
2		an32 646	. . . . . . . 8 ⊢ (((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3		disjlem14 38844	. . . . . . . . . . 11 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4		eleq2 2820	. . . . . . . . . . . 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 1934	. . 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 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056  dom cdm 5614  [cec 8620   Disj wdisjALTV 38257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-coss 38456  df-cnvrefrel 38572  df-disjALTV 38751

This theorem is referenced by:  disjlem18  38846