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Theorem funeldmdif 7981
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 6519 . . 3 (Fun 𝐴 → Rel 𝐴)
2 releldmdifi 7978 . . 3 ((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
31, 2sylan 581 . 2 ((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
4 eldif 3921 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
5 1stdm 7973 . . . . . . . . . . . . . 14 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
65ex 414 . . . . . . . . . . . . 13 (Rel 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
71, 6syl 17 . . . . . . . . . . . 12 (Fun 𝐴 → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
87adantr 482 . . . . . . . . . . 11 ((Fun 𝐴𝐵𝐴) → (𝑥𝐴 → (1st𝑥) ∈ dom 𝐴))
98com12 32 . . . . . . . . . 10 (𝑥𝐴 → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
109adantr 482 . . . . . . . . 9 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → ((Fun 𝐴𝐵𝐴) → (1st𝑥) ∈ dom 𝐴))
1110impcom 409 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → (1st𝑥) ∈ dom 𝐴)
12 funelss 7980 . . . . . . . . . . 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 456 . . . . . . . 8 (((Fun 𝐴𝐵𝐴) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐵)) → ¬ (1st𝑥) ∈ dom 𝐵)
1611, 15eldifd 3922 . . . . . . 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 2826 . . . . . . 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 231 . . . . 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 241 . . 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 211 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3074  cdif 3908  wss 3911  dom cdm 5634  Rel wrel 5639  Fun wfun 6491  cfv 6497  1st c1st 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  satffunlem2lem2  34003
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