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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 1912 . . . . . 6 𝑥 𝑦 ∈ (𝐹𝐴)
64, 5nfan 1897 . . . . 5 𝑥(𝜑𝑦 ∈ (𝐹𝐴))
7 nfv 1912 . . . . 5 𝑥 𝑦𝐵
8 id 22 . . . . . . . . . 10 ((𝐹𝑥) = 𝑦 → (𝐹𝑥) = 𝑦)
98eqcomd 2741 . . . . . . . . 9 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
1093ad2ant3 1134 . . . . . . . 8 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦 = (𝐹𝑥))
11 funimassd.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
12113adant3 1131 . . . . . . . 8 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
1310, 12eqeltrd 2839 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14133exp 1118 . . . . . 6 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
1514adantr 480 . . . . 5 ((𝜑𝑦 ∈ (𝐹𝐴)) → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
166, 7, 15rexlimd 3264 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
173, 16mpd 15 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦𝐵)
1817ex 412 . 2 (𝜑 → (𝑦 ∈ (𝐹𝐴) → 𝑦𝐵))
1918ssrdv 4001 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wnf 1780  wcel 2106  wrex 3068  wss 3963  cima 5692  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  ig1pmindeg  33602  aks6d1c3  42105  aks6d1c2lem4  42109  aks6d1c2  42112  aks6d1c6lem2  42153  funimaeq  45191
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