Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funelss Structured version   Visualization version   GIF version

Theorem funelss 7720
 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 6344 . . . . . 6 (Fun 𝐴 → Rel 𝐴)
2 1st2nd 7712 . . . . . 6 ((Rel 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
31, 2sylan 582 . . . . 5 ((Fun 𝐴𝑋𝐴) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
4 simpl1l 1220 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → Fun 𝐴)
5 simpl3 1189 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → 𝐵𝐴)
6 simpr 487 . . . . . . . . . 10 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (1st𝑋) ∈ dom 𝐵)
7 funssfv 6663 . . . . . . . . . 10 ((Fun 𝐴𝐵𝐴 ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (𝐵‘(1st𝑋)))
84, 5, 6, 7syl3anc 1367 . . . . . . . . 9 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (𝐵‘(1st𝑋)))
9 eleq1 2898 . . . . . . . . . . . . . . 15 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐴 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴))
109adantl 484 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐴 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴))
11 funopfv 6689 . . . . . . . . . . . . . . 15 (Fun 𝐴 → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1211adantr 483 . . . . . . . . . . . . . 14 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1310, 12sylbid 242 . . . . . . . . . . . . 13 ((Fun 𝐴𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝑋𝐴 → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1413impancom 454 . . . . . . . . . . . 12 ((Fun 𝐴𝑋𝐴) → (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝐴‘(1st𝑋)) = (2nd𝑋)))
1514imp 409 . . . . . . . . . . 11 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩) → (𝐴‘(1st𝑋)) = (2nd𝑋))
16153adant3 1128 . . . . . . . . . 10 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → (𝐴‘(1st𝑋)) = (2nd𝑋))
1716adantr 483 . . . . . . . . 9 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐴‘(1st𝑋)) = (2nd𝑋))
188, 17eqtr3d 2857 . . . . . . . 8 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝐵‘(1st𝑋)) = (2nd𝑋))
19 funss 6346 . . . . . . . . . . . . . 14 (𝐵𝐴 → (Fun 𝐴 → Fun 𝐵))
2019com12 32 . . . . . . . . . . . . 13 (Fun 𝐴 → (𝐵𝐴 → Fun 𝐵))
2120adantr 483 . . . . . . . . . . . 12 ((Fun 𝐴𝑋𝐴) → (𝐵𝐴 → Fun 𝐵))
2221imp 409 . . . . . . . . . . 11 (((Fun 𝐴𝑋𝐴) ∧ 𝐵𝐴) → Fun 𝐵)
2322funfnd 6358 . . . . . . . . . 10 (((Fun 𝐴𝑋𝐴) ∧ 𝐵𝐴) → 𝐵 Fn dom 𝐵)
24233adant2 1127 . . . . . . . . 9 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → 𝐵 Fn dom 𝐵)
25 fnopfvb 6691 . . . . . . . . 9 ((𝐵 Fn dom 𝐵 ∧ (1st𝑋) ∈ dom 𝐵) → ((𝐵‘(1st𝑋)) = (2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
2624, 25sylan 582 . . . . . . . 8 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → ((𝐵‘(1st𝑋)) = (2nd𝑋) ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
2718, 26mpbid 234 . . . . . . 7 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵)
28 eleq1 2898 . . . . . . . . 9 (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
29283ad2ant2 1130 . . . . . . . 8 (((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
3029adantr 483 . . . . . . 7 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → (𝑋𝐵 ↔ ⟨(1st𝑋), (2nd𝑋)⟩ ∈ 𝐵))
3127, 30mpbird 259 . . . . . 6 ((((Fun 𝐴𝑋𝐴) ∧ 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ 𝐵𝐴) ∧ (1st𝑋) ∈ dom 𝐵) → 𝑋𝐵)
32313exp1 1348 . . . . 5 ((Fun 𝐴𝑋𝐴) → (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
333, 32mpd 15 . . . 4 ((Fun 𝐴𝑋𝐴) → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵)))
3433ex 415 . . 3 (Fun 𝐴 → (𝑋𝐴 → (𝐵𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
3534com23 86 . 2 (Fun 𝐴 → (𝐵𝐴 → (𝑋𝐴 → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))))
36353imp 1107 1 ((Fun 𝐴𝐵𝐴𝑋𝐴) → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3909  ⟨cop 4545  dom cdm 5527  Rel wrel 5532  Fun wfun 6321   Fn wfn 6322  ‘cfv 6327  1st c1st 7661  2nd c2nd 7662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-iota 6286  df-fun 6329  df-fn 6330  df-fv 6335  df-1st 7663  df-2nd 7664 This theorem is referenced by:  funeldmdif  7721
 Copyright terms: Public domain W3C validator