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Theorem funimassd 38941
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)
21adantr 481 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → Fun 𝐹)
3 simpr 477 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦 ∈ (𝐹𝐴))
4 fvelima 6215 . . . 4 ((Fun 𝐹𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
52, 3, 4syl2anc 692 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
6 funimassd.1 . . . . 5 𝑥𝜑
7 nfv 1840 . . . . 5 𝑥 𝑦 ∈ (𝐹𝐴)
86, 7nfan 1825 . . . 4 𝑥(𝜑𝑦 ∈ (𝐹𝐴))
9 nfv 1840 . . . 4 𝑥 𝑦𝐵
10 id 22 . . . . . . . . 9 ((𝐹𝑥) = 𝑦 → (𝐹𝑥) = 𝑦)
1110eqcomd 2627 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
12113ad2ant3 1082 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦 = (𝐹𝑥))
13 funimassd.3 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
14133adant3 1079 . . . . . . 7 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
1512, 14eqeltrd 2698 . . . . . 6 ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
16153exp 1261 . . . . 5 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
1716adantr 481 . . . 4 ((𝜑𝑦 ∈ (𝐹𝐴)) → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
188, 9, 17rexlimd 3021 . . 3 ((𝜑𝑦 ∈ (𝐹𝐴)) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
195, 18mpd 15 . 2 ((𝜑𝑦 ∈ (𝐹𝐴)) → 𝑦𝐵)
2019ssd 38774 1 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987  wrex 2909  wss 3560  cima 5087  Fun wfun 5851  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by: (None)
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