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Theorem funeldmdif 8072
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 6585 . . 3 (Fun 𝐴 → Rel 𝐴)
2 releldmdifi 8069 . . 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 3973 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 1stdm 8064 . . . . . . . . . . . . . 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 8071 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴𝑥𝐴) → ((1st𝑥) ∈ dom 𝐵𝑥𝐵))
13123expa 1117 . . . . . . . . . 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 3974 . . . . . . 7 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
17163adant3 1131 . . . . . 6 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (1st𝑥) = 𝐶) → (1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵))
18 eleq1 2827 . . . . . . 7 ((1st𝑥) = 𝐶 → ((1st𝑥) ∈ (dom 𝐴 ∖ dom 𝐵) ↔ 𝐶 ∈ (dom 𝐴 ∖ dom 𝐵)))
19183ad2ant3 1134 . . . . . 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 1118 . . . 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 3151 . 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 1086   = wceq 1537  wcel 2106  wrex 3068  cdif 3960  wss 3963  dom cdm 5689  Rel wrel 5694  Fun wfun 6557  cfv 6563  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  satffunlem2lem2  35391
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