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Theorem funimaeq 43940
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 6579 . . . . . 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 7234 . . . . 5 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
116, 8, 9, 10syl3anc 1371 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
123, 11eqeltrd 2833 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
131, 2, 12funimassd 6958 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
142funfnd 6579 . . . . . 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 7234 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
1915, 17, 9, 18syl3anc 1371 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
203, 19eqeltrrd 2834 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
211, 4, 20funimassd 6958 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2213, 21eqssd 3999 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wss 3948  dom cdm 5676  cima 5679  Fun wfun 6537   Fn wfn 6538  cfv 6543
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by: (None)
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