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Theorem releldmdifi 7729
 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 3891 . . 3 (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ (𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵))
2 releldm2 7727 . . . . 5 (Rel 𝐴 → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
32adantr 484 . . . 4 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ dom 𝐴 ↔ ∃𝑥𝐴 (1st𝑥) = 𝐶))
43anbi1d 632 . . 3 ((Rel 𝐴𝐵𝐴) → ((𝐶 ∈ dom 𝐴 ∧ ¬ 𝐶 ∈ dom 𝐵) ↔ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)))
51, 4syl5bb 286 . 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 5621 . . . . . . . . . . . 12 (𝐵𝐴 → (Rel 𝐴 → Rel 𝐵))
87impcom 411 . . . . . . . . . . 11 ((Rel 𝐴𝐵𝐴) → Rel 𝐵)
9 1stdm 7724 . . . . . . . . . . 11 ((Rel 𝐵𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
108, 9sylan 583 . . . . . . . . . 10 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → (1st𝑥) ∈ dom 𝐵)
11 eleq1 2877 . . . . . . . . . 10 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ dom 𝐵𝐶 ∈ dom 𝐵))
1210, 11syl5ibcom 248 . . . . . . . . 9 (((Rel 𝐴𝐵𝐴) ∧ 𝑥𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1312rexlimdva 3243 . . . . . . . 8 ((Rel 𝐴𝐵𝐴) → (∃𝑥𝐵 (1st𝑥) = 𝐶𝐶 ∈ dom 𝐵))
1413con3d 155 . . . . . . 7 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶))
15 ralnex 3199 . . . . . . 7 (∀𝑥𝐵 ¬ (1st𝑥) = 𝐶 ↔ ¬ ∃𝑥𝐵 (1st𝑥) = 𝐶)
1614, 15syl6ibr 255 . . . . . 6 ((Rel 𝐴𝐵𝐴) → (¬ 𝐶 ∈ dom 𝐵 → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1716adantld 494 . . . . 5 ((Rel 𝐴𝐵𝐴) → ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶))
1817imp 410 . . . 4 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶)
19 rexdifi 4073 . . . 4 ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ∀𝑥𝐵 ¬ (1st𝑥) = 𝐶) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
206, 18, 19syl2anc 587 . . 3 (((Rel 𝐴𝐵𝐴) ∧ (∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵)) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶)
2120ex 416 . 2 ((Rel 𝐴𝐵𝐴) → ((∃𝑥𝐴 (1st𝑥) = 𝐶 ∧ ¬ 𝐶 ∈ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
225, 21sylbid 243 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  dom cdm 5520  Rel wrel 5525  ‘cfv 6325  1st c1st 7672 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fv 6333  df-1st 7674  df-2nd 7675 This theorem is referenced by:  funeldmdif  7732  satffunlem2lem2  32781
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