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Theorem funimassd 41719
 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 . . . 4 (𝜑 → Fun 𝐹)
2 fvelima 6714 . . . 4 ((Fun 𝐹𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
31, 2sylan 583 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
4 funimassd.1 . . . . 5 𝑥𝜑
5 nfv 1916 . . . . 5 𝑥 𝑦 ∈ (𝐹𝐴)
64, 5nfan 1901 . . . 4 𝑥(𝜑𝑦 ∈ (𝐹𝐴))
7 nfv 1916 . . . 4 𝑥 𝑦𝐵
8 id 22 . . . . . . . . 9 ((𝐹𝑥) = 𝑦 → (𝐹𝑥) = 𝑦)
98eqcomd 2830 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
1093ad2ant3 1132 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦 = (𝐹𝑥))
11 funimassd.3 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
12113adant3 1129 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
1310, 12eqeltrd 2916 . . . . . 6 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14133exp 1116 . . . . 5 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
1514adantr 484 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
166, 7, 15rexlimd 3309 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
173, 16mpd 15 . 2 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦𝐵)
1817ssd 41567 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2115  ∃wrex 3133   ⊆ wss 3918   “ cima 5541  Fun wfun 6332  ‘cfv 6338 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fv 6346 This theorem is referenced by:  funimaeq  41740
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