Theorem funimaeq 40260

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	. . . 4 ⊢ Ⅎ𝑥𝜑
2		funimaeq.e	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
3		funimaeq.g	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐺)
4	3	funfnd 6153	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn dom 𝐺)
5	4	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺)
6		funimaeq.d	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
7	6	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺)
8		simpr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
9		fnfvima 6751	. . . . . 6 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
10	5, 7, 8, 9	syl3anc 1496	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
11	2, 10	eqeltrd 2905	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
12	1, 11	ralrimia 40128	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
13		funimaeq.f	. . . 4 ⊢ (𝜑 → Fun 𝐹)
14		funimaeq.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
15		funimass4 6493	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)))
16	13, 14, 15	syl2anc 581	. . 3 ⊢ (𝜑 → ((𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)))
17	12, 16	mpbird 249	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴))
18	2	eqcomd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
19	13	funfnd 6153	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
20	19	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
21	14	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹)
22		fnfvima 6751	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
23	20, 21, 8, 22	syl3anc 1496	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
24	18, 23	eqeltrd 2905	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
25	1, 24	ralrimia 40128	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
26		funimass4 6493	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺) → ((𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)))
27	3, 6, 26	syl2anc 581	. . 3 ⊢ (𝜑 → ((𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)))
28	25, 27	mpbird 249	. 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴))
29	17, 28	eqssd 3843	1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658  Ⅎwnf 1884   ∈ wcel 2166  ∀wral 3116   ⊆ wss 3797  dom cdm 5341   “ cima 5344  Fun wfun 6116   Fn wfn 6117  ‘cfv 6122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-fv 6130

This theorem is referenced by: (None)