Theorem funimass4f 32894

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)

Hypotheses

Ref	Expression
funimass4f.1	⊢ Ⅎ𝑥𝐴
funimass4f.2	⊢ Ⅎ𝑥𝐵
funimass4f.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
funimass4f	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Proof of Theorem funimass4f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4f.3	. . . . . 6 ⊢ Ⅎ𝑥𝐹
2	1	nffun 6548	. . . . 5 ⊢ Ⅎ𝑥Fun 𝐹
3		funimass4f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	1	nfdm 5932	. . . . . 6 ⊢ Ⅎ𝑥dom 𝐹
5	3, 4	nfss 3932	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ dom 𝐹
6	2, 5	nfan 1922	. . . 4 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)
7	1, 3	nfima 6061	. . . . 5 ⊢ Ⅎ𝑥(𝐹 “ 𝐴)
8		funimass4f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
9	7, 8	nfss 3932	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) ⊆ 𝐵
10	6, 9	nfan 1922	. . 3 ⊢ Ⅎ𝑥((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵)
11		funfvima2 7219	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
12		ssel 3933	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ 𝐵))
13	11, 12	sylan9 516	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵))
14	10, 13	ralrimi 3263	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
15	3, 1	dfimafnf 32893	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
16	15	adantr 485	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
17	8	abrexss 32768	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
18	17	adantl 486	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
19	16, 18	eqsstrd 3973	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) ⊆ 𝐵)
20	14, 19	impbida 812	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  Ⅎwnfc 2912  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  dom cdm 5652   “ cima 5655  Fun wfun 6519  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533

This theorem is referenced by:  ballotlem7  34843