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Theorem funimaeq 44504
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 6572 . . . . . 6 (𝜑𝐺 Fn dom 𝐺)
65adantr 480 . . . . 5 ((𝜑𝑥𝐴) → 𝐺 Fn dom 𝐺)
7 funimaeq.d . . . . . 6 (𝜑𝐴 ⊆ dom 𝐺)
87adantr 480 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐺)
9 simpr 484 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐴)
10 fnfvima 7229 . . . . 5 ((𝐺 Fn dom 𝐺𝐴 ⊆ dom 𝐺𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
116, 8, 9, 10syl3anc 1368 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐺𝐴))
123, 11eqeltrd 2827 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐺𝐴))
131, 2, 12funimassd 6951 . 2 (𝜑 → (𝐹𝐴) ⊆ (𝐺𝐴))
142funfnd 6572 . . . . . 6 (𝜑𝐹 Fn dom 𝐹)
1514adantr 480 . . . . 5 ((𝜑𝑥𝐴) → 𝐹 Fn dom 𝐹)
16 funimaeq.a . . . . . 6 (𝜑𝐴 ⊆ dom 𝐹)
1716adantr 480 . . . . 5 ((𝜑𝑥𝐴) → 𝐴 ⊆ dom 𝐹)
18 fnfvima 7229 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ⊆ dom 𝐹𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
1915, 17, 9, 18syl3anc 1368 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
203, 19eqeltrrd 2828 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ (𝐹𝐴))
211, 4, 20funimassd 6951 . 2 (𝜑 → (𝐺𝐴) ⊆ (𝐹𝐴))
2213, 21eqssd 3994 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wnf 1777  wcel 2098  wss 3943  dom cdm 5669  cima 5672  Fun wfun 6530   Fn wfn 6531  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544
This theorem is referenced by: (None)
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