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Theorem funimaeq 41394
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
funimaeq.x 𝑥𝜑
funimaeq.f (𝜑 → Fun 𝐹)
funimaeq.g (𝜑 → Fun 𝐺)
funimaeq.a (𝜑𝐴 ⊆ dom 𝐹)
funimaeq.d (𝜑𝐴 ⊆ dom 𝐺)
funimaeq.e ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
Assertion
Ref Expression
funimaeq (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem funimaeq
StepHypRef Expression
1 funimaeq.x . . 3 𝑥𝜑
2 funimaeq.f . . 3 (𝜑 → Fun 𝐹)
3 funimaeq.e . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
4 funimaeq.g . . . . . . 7 (𝜑 → Fun 𝐺)
54funfnd 6379 . . . . . 6 (𝜑𝐺 Fn dom 𝐺)
65adantr 481 . . . . 5 ((𝜑𝑥𝐴) → 𝐺 Fn dom 𝐺)
7 funimaeq.d . . . . . 6 (𝜑𝐴 ⊆ dom 𝐺)
87adantr 481 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐺)
9 simpr 485 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐴)
10 fnfvima 6986 . . . . 5 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
116, 8, 9, 10syl3anc 1363 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
123, 11eqeltrd 2910 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
131, 2, 12funimassd 41373 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
142funfnd 6379 . . . . . 6 (𝜑𝐹 Fn dom 𝐹)
1514adantr 481 . . . . 5 ((𝜑𝑥𝐴) → 𝐹 Fn dom 𝐹)
16 funimaeq.a . . . . . 6 (𝜑𝐴 ⊆ dom 𝐹)
1716adantr 481 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐹)
18 fnfvima 6986 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
1915, 17, 9, 18syl3anc 1363 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
203, 19eqeltrrd 2911 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
211, 4, 20funimassd 41373 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2213, 21eqssd 3981 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wnf 1775  wcel 2105  wss 3933  dom cdm 5548  cima 5551  Fun wfun 6342   Fn wfn 6343  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-fn 6351  df-fv 6356
This theorem is referenced by: (None)
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