Theorem funimass4f 32697

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 6516	. . . . 5 ⊢ Ⅎ𝑥Fun 𝐹
3		funimass4f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
4	1	nfdm 5901	. . . . . 6 ⊢ Ⅎ𝑥dom 𝐹
5	3, 4	nfss 3927	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ dom 𝐹
6	2, 5	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹)
7	1, 3	nfima 6028	. . . . 5 ⊢ Ⅎ𝑥(𝐹 “ 𝐴)
8		funimass4f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
9	7, 8	nfss 3927	. . . 4 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) ⊆ 𝐵
10	6, 9	nfan 1901	. . 3 ⊢ Ⅎ𝑥((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵)
11		funfvima2 7179	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
12		ssel 3928	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ 𝐵))
13	11, 12	sylan9 507	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵))
14	10, 13	ralrimi 3235	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
15	3, 1	dfimafnf 32696	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
16	15	adantr 480	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
17	8	abrexss 32569	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
18	17	adantl 481	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵)
19	16, 18	eqsstrd 3969	. 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) ⊆ 𝐵)
20	14, 19	impbida 801	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902  dom cdm 5625   “ cima 5628  Fun wfun 6487  ‘cfv 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by:  ballotlem7  34674