Theorem funimaeq 45240

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	⊢ Ⅎ𝑥𝜑
funimaeq.f	⊢ (𝜑 → Fun 𝐹)
funimaeq.g	⊢ (𝜑 → Fun 𝐺)
funimaeq.a	⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
funimaeq.d	⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
funimaeq.e	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))

Assertion

Ref	Expression
funimaeq	⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem funimaeq

Step	Hyp	Ref	Expression
1		funimaeq.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		funimaeq.f	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		funimaeq.e	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
4		funimaeq.g	. . . . . . 7 ⊢ (𝜑 → Fun 𝐺)
5	4	funfnd 6547	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺)
7		funimaeq.d	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺)
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
10		fnfvima 7207	. . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
11	6, 8, 9, 10	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
12	3, 11	eqeltrd 2828	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
13	1, 2, 12	funimassd 6927	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴))
14	2	funfnd 6547	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
16		funimaeq.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹)
18		fnfvima 7207	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
19	15, 17, 9, 18	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
20	3, 19	eqeltrrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
21	1, 4, 20	funimassd 6927	. 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴))
22	13, 21	eqssd 3964	1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109   ⊆ wss 3914  dom cdm 5638   “ cima 5641  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by: (None)