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Theorem funelss 7980
Description: If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.)
Assertion
Ref Expression
funelss ((Fun 𝐴𝐵𝐴𝑋𝐴) → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))

Proof of Theorem funelss
StepHypRef Expression
1 funrel 6519 . . . . . 6 (Fun 𝐴 → Rel 𝐴)
2 1st2nd 7972 . . . . . 6 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 581 . . . . 5 ((Fun 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 simpl1l 1225 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → Fun 𝐴)
5 simpl3 1194 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → 𝐵𝐴)
6 simpr 486 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (1st𝑋) ∈ dom 𝐵)
7 funssfv 6864 . . . . . . . . . 10 ((Fun 𝐴𝐵𝐴 ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (𝐵‘(1st𝑋)))
84, 5, 6, 7syl3anc 1372 . . . . . . . . 9 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (𝐵‘(1st𝑋)))
9 eleq1 2826 . . . . . . . . . . . . . . 15 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐴 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴))
109adantl 483 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐴 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴))
11 funopfv 6895 . . . . . . . . . . . . . . 15 (Fun 𝐴 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1211adantr 482 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1310, 12sylbid 239 . . . . . . . . . . . . 13 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1413impancom 453 . . . . . . . . . . . 12 ((Fun 𝐴𝑋𝐴) → (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1514imp 408 . . . . . . . . . . 11 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝐴‘(1st𝑋)) = (2nd𝑋))
16153adant3 1133 . . . . . . . . . 10 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → (𝐴‘(1st𝑋)) = (2nd𝑋))
1716adantr 482 . . . . . . . . 9 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (2nd𝑋))
188, 17eqtr3d 2779 . . . . . . . 8 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐵‘(1st𝑋)) = (2nd𝑋))
19 funss 6521 . . . . . . . . . . . . . 14 (𝐵𝐴 → (Fun 𝐴 → Fun 𝐵))
2019com12 32 . . . . . . . . . . . . 13 (Fun 𝐴 → (𝐵𝐴 → Fun 𝐵))
2120adantr 482 . . . . . . . . . . . 12 ((Fun 𝐴𝑋𝐴) → (𝐵𝐴 → Fun 𝐵))
2221imp 408 . . . . . . . . . . 11 (((Fun 𝐴𝑋𝐴) ∧ 𝐵𝐴) → Fun 𝐵)
2322funfnd 6533 . . . . . . . . . 10 (((Fun 𝐴𝑋𝐴) ∧ 𝐵𝐴) → 𝐵 Fn dom 𝐵)
24233adant2 1132 . . . . . . . . 9 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → 𝐵 Fn dom 𝐵)
25 fnopfvb 6897 . . . . . . . . 9 ((𝐵 Fn dom 𝐵 ∧ (1st𝑋) ∈ dom 𝐵) → ((𝐵‘(1st𝑋)) = (2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
2624, 25sylan 581 . . . . . . . 8 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → ((𝐵‘(1st𝑋)) = (2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
2718, 26mpbid 231 . . . . . . 7 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵)
28 eleq1 2826 . . . . . . . . 9 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
29283ad2ant2 1135 . . . . . . . 8 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
3029adantr 482 . . . . . . 7 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
3127, 30mpbird 257 . . . . . 6 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → 𝑋𝐵)
32313exp1 1353 . . . . 5 ((Fun 𝐴𝑋𝐴) → (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
333, 32mpd 15 . . . 4 ((Fun 𝐴𝑋𝐴) → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵)))
3433ex 414 . . 3 (Fun 𝐴 → (𝑋𝐴 → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
3534com23 86 . 2 (Fun 𝐴 → (𝐵𝐴 → (𝑋𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
36353imp 1112 1 ((Fun 𝐴𝐵𝐴𝑋𝐴) → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3911  cop 4593  dom cdm 5634  Rel wrel 5639  Fun wfun 6491   Fn wfn 6492  cfv 6497  1st c1st 7920  2nd c2nd 7921
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-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:  funeldmdif  7981
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