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Theorem funimassd 41373
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 6724 . . . 4 ((Fun 𝐹𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
31, 2sylan 580 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
4 funimassd.1 . . . . 5 𝑥𝜑
5 nfv 1906 . . . . 5 𝑥 𝑦 ∈ (𝐹𝐴)
64, 5nfan 1891 . . . 4 𝑥(𝜑𝑦 ∈ (𝐹𝐴))
7 nfv 1906 . . . 4 𝑥 𝑦𝐵
8 id 22 . . . . . . . . 9 ((𝐹𝑥) = 𝑦 → (𝐹𝑥) = 𝑦)
98eqcomd 2824 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
1093ad2ant3 1127 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦 = (𝐹𝑥))
11 funimassd.3 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
12113adant3 1124 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
1310, 12eqeltrd 2910 . . . . . 6 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14133exp 1111 . . . . 5 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
1514adantr 481 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
166, 7, 15rexlimd 3314 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
173, 16mpd 15 . 2 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦𝐵)
1817ssd 41221 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wnf 1775  wcel 2105  wrex 3136  wss 3933  cima 5551  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  funimaeq  41394
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