Theorem funimaeq 46397

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 6579	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn dom 𝐺)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺)
7		funimaeq.d	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺)
9		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
10		fnfvima 7246	. . . . 5 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
11	6, 8, 9, 10	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
12	3, 11	eqeltrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
13	1, 2, 12	funimassd 6959	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴))
14	2	funfnd 6579	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
15	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
16		funimaeq.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
17	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹)
18		fnfvima 7246	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
19	15, 17, 9, 18	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
20	3, 19	eqeltrrd 2864	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
21	1, 4, 20	funimassd 6959	. 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴))
22	13, 21	eqssd 3929	1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2146   ⊆ wss 3899  dom cdm 5654   “ cima 5657  Fun wfun 6540   Fn wfn 6541  ‘cfv 6546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554

This theorem is used by: (None)