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Theorem disjlem17 38781
Description: Lemma for disjdmqseq 38787, partim2 38789 and petlem 38794 via disjlem18 38782, (general version of the former prtlem17 38858). (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 3069 . . 3 (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)))
2 an32 646 . . . . . . . 8 (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3 disjlem14 38780 . . . . . . . . . . 11 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4 eleq2 2828 . . . . . . . . . . . 12 ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅𝐵 ∈ [𝑦]𝑅))
54biimprd 248 . . . . . . . . . . 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 431 . . . . . . . . 9 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
87impd 410 . . . . . . . 8 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
92, 8biimtrrid 243 . . . . . . 7 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
109expd 415 . . . . . 6 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
1110imp5a 440 . . . . 5 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
1211imp4b 421 . . . 4 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
1312exlimdv 1931 . . 3 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
141, 13biimtrid 242 . 2 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
1514ex 412 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 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  dom cdm 5689  [cec 8742   Disj wdisjALTV 38196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-coss 38393  df-cnvrefrel 38509  df-disjALTV 38687
This theorem is referenced by:  disjlem18  38782
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