Theorem funimaeq 39461

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 5919	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn dom 𝐺)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn dom 𝐺)
6		funimaeq.d	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
7	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐺)
8		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
9		fnfvima 6496	. . . . . 6 ⊢ ((𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
10	5, 7, 8, 9	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐺 “ 𝐴))
11	2, 10	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐺 “ 𝐴))
12	1, 11	ralrimia 39315	. . 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 6247	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)))
16	13, 14, 15	syl2anc 693	. . 3 ⊢ (𝜑 → ((𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ (𝐺 “ 𝐴)))
17	12, 16	mpbird 247	. 2 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ (𝐺 “ 𝐴))
18	2	eqcomd 2628	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐹‘𝑥))
19	13	funfnd 5919	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
20	19	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn dom 𝐹)
21	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ dom 𝐹)
22		fnfvima 6496	. . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
23	20, 21, 8, 22	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))
24	18, 23	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
25	1, 24	ralrimia 39315	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴))
26		funimass4 6247	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ⊆ dom 𝐺) → ((𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)))
27	3, 6, 26	syl2anc 693	. . 3 ⊢ (𝜑 → ((𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹 “ 𝐴)))
28	25, 27	mpbird 247	. 2 ⊢ (𝜑 → (𝐺 “ 𝐴) ⊆ (𝐹 “ 𝐴))
29	17, 28	eqssd 3620	1 ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)