Theorem disjlem17 39398

Description: Lemma for disjdmqseq 39404, partim2 39406 and petlem 39411 via disjlem18 39399, (general version of the former prtlem17 39497). (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 656	. . . . . . . 8 ⊢ (((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3		disjlem14 39397	. . . . . . . . . . 11 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4		eleq2 2851	. . . . . . . . . . . 12 ⊢ ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅 ↔ 𝐵 ∈ [𝑦]𝑅))
5	4	biimprd 250	. . . . . . . . . . 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 435	. . . . . . . . 9 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
8	7	impd 414	. . . . . . . 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 245	. . . . . . 7 ⊢ ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅))))
10	9	expd 419	. . . . . 6 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅 → 𝐵 ∈ [𝑥]𝑅)))))
11	10	imp5a 444	. . . . 5 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
12	11	imp4b 425	. . . 4 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
13	12	exlimdv 1953	. . 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 244	. 2 ⊢ (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅 ∧ 𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅 ∧ 𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
15	14	ex 416	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 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∃wrex 3086  dom cdm 5647  [cec 8676   Disj wdisjALTV 38715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-coss 38997  df-cnvrefrel 39103  df-disjALTV 39286

This theorem is referenced by:  disjlem18  39399