Theorem funimaeq 44651

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 6589	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺)
6	5	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺)
7		funimaeq.d	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
8	7	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺)
9		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
10		fnfvima 7251	. . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
11	6, 8, 9, 10	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
12	3, 11	eqeltrd 2829	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
13	1, 2, 12	funimassd 6970	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴))
14	2	funfnd 6589	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
15	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
16		funimaeq.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
17	16	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹)
18		fnfvima 7251	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
19	15, 17, 9, 18	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
20	3, 19	eqeltrrd 2830	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
21	1, 4, 20	funimassd 6970	. 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴))
22	13, 21	eqssd 3999	1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098   ⊆ wss 3949  dom cdm 5682   “ cima 5685  Fun wfun 6547   Fn wfn 6548  ‘cfv 6553

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561

This theorem is referenced by: (None)