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Theorem funeldmdif 8073
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 6583 . . 3 (Fun 𝐴 → Rel 𝐴)
2 releldmdifi 8070 . . 3 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
31, 2sylan 580 . 2 ((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
4 eldif 3961 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 1stdm 8065 . . . . . . . . . . . . . 14 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
65ex 412 . . . . . . . . . . . . 13 (Rel 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
71, 6syl 17 . . . . . . . . . . . 12 (Fun 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
87adantr 480 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴) → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
98com12 32 . . . . . . . . . 10 (𝑥𝐴 → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
109adantr 480 . . . . . . . . 9 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
1110impcom 407 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ dom 𝐴)
12 funelss 8072 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴𝑥𝐴) → ((1st𝑥) ∈ dom 𝐵𝑥𝐵))
13123expa 1119 . . . . . . . . . 10 (((Fun 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ((1st𝑥) ∈ dom 𝐵𝑥𝐵))
1413con3d 152 . . . . . . . . 9 (((Fun 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ¬ (1st𝑥) ∈ dom 𝐵))
1514impr 454 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → ¬ (1st𝑥) ∈ dom 𝐵)
1611, 15eldifd 3962 . . . . . . 7 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17163adant3 1133 . . . . . 6 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18 eleq1 2829 . . . . . . 7 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19183ad2ant3 1136 . . . . . 6 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → ((1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
2017, 19mpbid 232 . . . . 5 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))
21203exp 1120 . . . 4 ((Fun 𝐴𝐵𝐴) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
224, 21biimtrid 242 . . 3 ((Fun 𝐴𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵))))
2322rexlimdv 3153 . 2 ((Fun 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
243, 23impbid 212 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  wss 3951  dom cdm 5685  Rel wrel 5690  Fun wfun 6555  cfv 6561  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  satffunlem2lem2  35411
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