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Theorem releldmdifi 8031
Description: One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)
Assertion
Ref Expression
releldmdifi ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem releldmdifi
StepHypRef Expression
1 eldif 3959 . . 3 (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵))
2 releldm2 8029 . . . . 5 (Rel 𝐴 → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
32adantr 482 . . . 4 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
43anbi1d 631 . . 3 ((Rel 𝐴𝐵𝐴) → ((𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵) ↔ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
51, 4bitrid 283 . 2 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
6 simprl 770 . . . 4 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥𝐴 (1st𝑥) = 𝐶)
7 relss 5782 . . . . . . . . . . . 12 (𝐵𝐴 → (Rel 𝐴 → Rel 𝐵))
87impcom 409 . . . . . . . . . . 11 ((Rel 𝐴𝐵𝐴) → Rel 𝐵)
9 1stdm 8026 . . . . . . . . . . 11 ((Rel 𝐵𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
108, 9sylan 581 . . . . . . . . . 10 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
11 eleq1 2822 . . . . . . . . . 10 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ dom 𝐵𝐶 ∈ dom 𝐵))
1210, 11syl5ibcom 244 . . . . . . . . 9 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1312rexlimdva 3156 . . . . . . . 8 ((Rel 𝐴𝐵𝐴) → (∃𝑥𝐵 (1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1413con3d 152 . . . . . . 7 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶))
15 ralnex 3073 . . . . . . 7 (∀𝑥𝐵 ¬ (1st𝑥) = 𝐶 ↔ ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶)
1614, 15imbitrrdi 251 . . . . . 6 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1716adantld 492 . . . . 5 ((Rel 𝐴𝐵𝐴) → ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1817imp 408 . . . 4 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶)
19 rexdifi 4146 . . . 4 ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
206, 18, 19syl2anc 585 . . 3 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
2120ex 414 . 2 ((Rel 𝐴𝐵𝐴) → ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
225, 21sylbid 239 1 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cdif 3946  wss 3949  dom cdm 5677  Rel wrel 5682  cfv 6544  1st c1st 7973
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  ax-un 7725
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-ne 2942  df-ral 3063  df-rex 3072  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-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  funeldmdif  8034  satffunlem2lem2  34397
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