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Theorem funimaeq 39979
 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 6081 . . . . . . 7 (𝜑𝐺 Fn dom 𝐺)
54adantr 472 . . . . . 6 ((𝜑𝑥𝐴) → 𝐺 Fn dom 𝐺)
6 funimaeq.d . . . . . . 7 (𝜑𝐴 ⊆ dom 𝐺)
76adantr 472 . . . . . 6 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐺)
8 simpr 479 . . . . . 6 ((𝜑𝑥𝐴) → 𝑥𝐴)
9 fnfvima 6661 . . . . . 6 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
105, 7, 8, 9syl3anc 1477 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
112, 10eqeltrd 2840 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
121, 11ralrimia 39833 . . 3 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴))
13 funimaeq.f . . . 4 (𝜑 → Fun 𝐹)
14 funimaeq.a . . . 4 (𝜑𝐴 ⊆ dom 𝐹)
15 funimass4 6411 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ (𝐺𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴)))
1613, 14, 15syl2anc 696 . . 3 (𝜑 → ((𝐹𝐴) ⊆ (𝐺𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐺𝐴)))
1712, 16mpbird 247 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
182eqcomd 2767 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐹𝑥))
1913funfnd 6081 . . . . . . 7 (𝜑𝐹 Fn dom 𝐹)
2019adantr 472 . . . . . 6 ((𝜑𝑥𝐴) → 𝐹 Fn dom 𝐹)
2114adantr 472 . . . . . 6 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐹)
22 fnfvima 6661 . . . . . 6 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2320, 21, 8, 22syl3anc 1477 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2418, 23eqeltrd 2840 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
251, 24ralrimia 39833 . . 3 (𝜑 → ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴))
26 funimass4 6411 . . . 4 ((Fun 𝐺𝐴 ⊆ dom 𝐺) → ((𝐺𝐴) ⊆ (𝐹𝐴) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴)))
273, 6, 26syl2anc 696 . . 3 (𝜑 → ((𝐺𝐴) ⊆ (𝐹𝐴) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝐴)))
2825, 27mpbird 247 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2917, 28eqssd 3762 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632  Ⅎwnf 1857   ∈ wcel 2140  ∀wral 3051   ⊆ wss 3716  dom cdm 5267   “ cima 5270  Fun wfun 6044   Fn wfn 6045  ‘cfv 6050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-fv 6058 This theorem is referenced by: (None)
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