Theorem funimass4f 30376

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 6372	. . . . 5 ⊢ Ⅎ𝑥Fun 𝐹
3		funimass4f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	1	nfdm 5817	. . . . . 6 ⊢ Ⅎ𝑥dom 𝐹
5	3, 4	nfss 3959	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ dom 𝐹
6	2, 5	nfan 1896	. . . 4 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)
7	1, 3	nfima 5931	. . . . 5 ⊢ Ⅎ𝑥(𝐹 “ 𝐴)
8		funimass4f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
9	7, 8	nfss 3959	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) ⊆ 𝐵
10	6, 9	nfan 1896	. . 3 ⊢ Ⅎ𝑥((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵)
11		funfvima2 6987	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
12		ssel 3960	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ 𝐵))
13	11, 12	sylan9 510	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵))
14	10, 13	ralrimi 3216	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
15	3, 1	dfimafnf 30375	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
16	15	adantr 483	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
17	8	abrexss 30266	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
18	17	adantl 484	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
19	16, 18	eqsstrd 4004	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) ⊆ 𝐵)
20	14, 19	impbida 799	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  Ⅎwnfc 2961  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  dom cdm 5549   “ cima 5552  Fun wfun 6343  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by:  ballotlem7  31788