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Theorem funeldmdif 7742
Description: Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)
Assertion
Ref Expression
funeldmdif ((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem funeldmdif
StepHypRef Expression
1 funrel 6360 . . 3 (Fun 𝐴 → Rel 𝐴)
2 releldmdifi 7739 . . 3 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
31, 2sylan 583 . 2 ((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
4 eldif 3929 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 1stdm 7734 . . . . . . . . . . . . . 14 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
65ex 416 . . . . . . . . . . . . 13 (Rel 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
71, 6syl 17 . . . . . . . . . . . 12 (Fun 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
87adantr 484 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴) → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
98com12 32 . . . . . . . . . 10 (𝑥𝐴 → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
109adantr 484 . . . . . . . . 9 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
1110impcom 411 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ dom 𝐴)
12 funelss 7741 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴𝑥𝐴) → ((1st𝑥) ∈ dom 𝐵𝑥𝐵))
13123expa 1115 . . . . . . . . . 10 (((Fun 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((1st𝑥) ∈ dom 𝐵𝑥𝐵))
1413con3d 155 . . . . . . . . 9 (((Fun 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ¬ (1st𝑥) ∈ dom 𝐵))
1514impr 458 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → ¬ (1st𝑥) ∈ dom 𝐵)
1611, 15eldifd 3930 . . . . . . 7 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17163adant3 1129 . . . . . 6 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18 eleq1 2903 . . . . . . 7 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19183ad2ant3 1132 . . . . . 6 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → ((1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
2017, 19mpbid 235 . . . . 5 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))
21203exp 1116 . . . 4 ((Fun 𝐴𝐵𝐴) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
224, 21syl5bi 245 . . 3 ((Fun 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
2322rexlimdv 3275 . 2 ((Fun 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
243, 23impbid 215 1 ((Fun 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  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  cdif 3916  wss 3919  dom cdm 5542  Rel wrel 5547  Fun wfun 6337  cfv 6343  1st c1st 7682
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-1st 7684  df-2nd 7685
This theorem is referenced by:  satffunlem2lem2  32710
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