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Theorem funimaeq 40260
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 . . . 4 𝑥𝜑
2 funimaeq.e . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
3 funimaeq.g . . . . . . . 8 (𝜑 → Fun 𝐺)
43funfnd 6153 . . . . . . 7 (𝜑𝐺 Fn dom 𝐺)
54adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐺 Fn dom 𝐺)
6 funimaeq.d . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐺)
76adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐺)
8 simpr 479 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
9 fnfvima 6751 . . . . . 6 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
105, 7, 8, 9syl3anc 1496 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
112, 10eqeltrd 2905 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
121, 11ralrimia 40128 . . 3 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴))
13 funimaeq.f . . . 4 (𝜑 → Fun 𝐹)
14 funimaeq.a . . . 4 (𝜑𝐴 ⊆ dom 𝐹)
15 funimass4 6493 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ (𝐺𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴)))
1613, 14, 15syl2anc 581 . . 3 (𝜑 → ((𝐹𝐴) ⊆ (𝐺𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴)))
1712, 16mpbird 249 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
182eqcomd 2830 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
1913funfnd 6153 . . . . . . 7 (𝜑𝐹 Fn dom 𝐹)
2019adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐹 Fn dom 𝐹)
2114adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐹)
22 fnfvima 6751 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2320, 21, 8, 22syl3anc 1496 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2418, 23eqeltrd 2905 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
251, 24ralrimia 40128 . . 3 (𝜑 → ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴))
26 funimass4 6493 . . . 4 ((Fun 𝐺𝐴 ⊆ dom 𝐺) → ((𝐺𝐴) ⊆ (𝐹𝐴) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴)))
273, 6, 26syl2anc 581 . . 3 (𝜑 → ((𝐺𝐴) ⊆ (𝐹𝐴) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴)))
2825, 27mpbird 249 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2917, 28eqssd 3843 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wnf 1884  wcel 2166  wral 3116  wss 3797  dom cdm 5341  cima 5344  Fun wfun 6116   Fn wfn 6117  cfv 6122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-fv 6130
This theorem is referenced by: (None)
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