MPE Home 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