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Theorem disjlem17 37669
Description: Lemma for disjdmqseq 37675, partim2 37677 and petlem 37682 via disjlem18 37670, (general version of the former prtlem17 37746). (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
disjlem17 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [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
StepHypRef Expression
1 df-rex 3072 . . 3 (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)))
2 an32 645 . . . . . . . 8 (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3 disjlem14 37668 . . . . . . . . . . 11 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4 eleq2 2823 . . . . . . . . . . . 12 ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅𝐵 ∈ [𝑦]𝑅))
54biimprd 247 . . . . . . . . . . 11 ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))
63, 5syl8 76 . . . . . . . . . 10 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
76exp4a 433 . . . . . . . . 9 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
87impd 412 . . . . . . . 8 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
92, 8biimtrrid 242 . . . . . . 7 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
109expd 417 . . . . . 6 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
1110imp5a 442 . . . . 5 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
1211imp4b 423 . . . 4 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
1312exlimdv 1937 . . 3 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
141, 13biimtrid 241 . 2 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
1514ex 414 1 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  dom cdm 5677  [cec 8701   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:  disjlem18  37670
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