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Theorem funimaeq 41809
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 6374 . . . . . 6 (𝜑𝐺 Fn dom 𝐺)
65adantr 484 . . . . 5 ((𝜑𝑥𝐴) → 𝐺 Fn dom 𝐺)
7 funimaeq.d . . . . . 6 (𝜑𝐴 ⊆ dom 𝐺)
87adantr 484 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐺)
9 simpr 488 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐴)
10 fnfvima 6987 . . . . 5 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
116, 8, 9, 10syl3anc 1368 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
123, 11eqeltrd 2916 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
131, 2, 12funimassd 41788 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
142funfnd 6374 . . . . . 6 (𝜑𝐹 Fn dom 𝐹)
1514adantr 484 . . . . 5 ((𝜑𝑥𝐴) → 𝐹 Fn dom 𝐹)
16 funimaeq.a . . . . . 6 (𝜑𝐴 ⊆ dom 𝐹)
1716adantr 484 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐹)
18 fnfvima 6987 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
1915, 17, 9, 18syl3anc 1368 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
203, 19eqeltrrd 2917 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
211, 4, 20funimassd 41788 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2213, 21eqssd 3970 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2115  wss 3919  dom cdm 5542  cima 5545  Fun wfun 6337   Fn wfn 6338  cfv 6343
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 5189  ax-nul 5196  ax-pr 5317
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 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351
This theorem is referenced by: (None)
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