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

Theorem funimassd 6975
Description: Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
funimassd.1 𝑥𝜑
funimassd.2 (𝜑 → Fun 𝐹)
funimassd.3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
Assertion
Ref Expression
funimassd (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem funimassd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funimassd.2 . . . . 5 (𝜑 → Fun 𝐹)
2 fvelima 6974 . . . . 5 ((Fun 𝐹𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
31, 2sylan 580 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
4 funimassd.1 . . . . . 6 𝑥𝜑
5 nfv 1914 . . . . . 6 𝑥 𝑦 ∈ (𝐹𝐴)
64, 5nfan 1899 . . . . 5 𝑥(𝜑𝑦 ∈ (𝐹𝐴))
7 nfv 1914 . . . . 5 𝑥 𝑦𝐵
8 id 22 . . . . . . . . . 10 ((𝐹𝑥) = 𝑦 → (𝐹𝑥) = 𝑦)
98eqcomd 2743 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
1093ad2ant3 1136 . . . . . . . 8 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦 = (𝐹𝑥))
11 funimassd.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
12113adant3 1133 . . . . . . . 8 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
1310, 12eqeltrd 2841 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14133exp 1120 . . . . . 6 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
1514adantr 480 . . . . 5 ((𝜑𝑦 ∈ (𝐹𝐴)) → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
166, 7, 15rexlimd 3266 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
173, 16mpd 15 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦𝐵)
1817ex 412 . 2 (𝜑 → (𝑦 ∈ (𝐹𝐴) → 𝑦𝐵))
1918ssrdv 3989 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wnf 1783  wcel 2108  wrex 3070  wss 3951  cima 5688  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  ig1pmindeg  33622  exsslsb  33647  aks6d1c3  42124  aks6d1c2lem4  42128  aks6d1c2  42131  aks6d1c6lem2  42172  funimaeq  45253
  Copyright terms: Public domain W3C validator