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Theorem releldmdifi 8033
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 3958 . . 3 (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵))
2 releldm2 8031 . . . . 5 (Rel 𝐴 → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
32adantr 481 . . . 4 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
43anbi1d 630 . . 3 ((Rel 𝐴𝐵𝐴) → ((𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵) ↔ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
51, 4bitrid 282 . 2 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
6 simprl 769 . . . 4 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥𝐴 (1st𝑥) = 𝐶)
7 relss 5781 . . . . . . . . . . . 12 (𝐵𝐴 → (Rel 𝐴 → Rel 𝐵))
87impcom 408 . . . . . . . . . . 11 ((Rel 𝐴𝐵𝐴) → Rel 𝐵)
9 1stdm 8028 . . . . . . . . . . 11 ((Rel 𝐵𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
108, 9sylan 580 . . . . . . . . . 10 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
11 eleq1 2821 . . . . . . . . . 10 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ dom 𝐵𝐶 ∈ dom 𝐵))
1210, 11syl5ibcom 244 . . . . . . . . 9 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1312rexlimdva 3155 . . . . . . . 8 ((Rel 𝐴𝐵𝐴) → (∃𝑥𝐵 (1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1413con3d 152 . . . . . . 7 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶))
15 ralnex 3072 . . . . . . 7 (∀𝑥𝐵 ¬ (1st𝑥) = 𝐶 ↔ ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶)
1614, 15imbitrrdi 251 . . . . . 6 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1716adantld 491 . . . . 5 ((Rel 𝐴𝐵𝐴) → ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1817imp 407 . . . 4 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶)
19 rexdifi 4145 . . . 4 ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
206, 18, 19syl2anc 584 . . 3 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
2120ex 413 . 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 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cdif 3945  wss 3948  dom cdm 5676  Rel wrel 5681  cfv 6543  1st c1st 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7977  df-2nd 7978
This theorem is referenced by:  funeldmdif  8036  satffunlem2lem2  34466
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