Theorem funimassd 42659

Description: Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
funimassd.1	⊢ Ⅎ𝑥𝜑
funimassd.2	⊢ (𝜑 → Fun 𝐹)
funimassd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)

Assertion

Ref	Expression
funimassd	⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem funimassd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimassd.2	. . . 4 ⊢ (𝜑 → Fun 𝐹)
2		fvelima 6817	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
3	1, 2	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
4		funimassd.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
5		nfv 1918	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴)
6	4, 5	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴))
7		nfv 1918	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
8		id 22	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦)
9	8	eqcomd 2744	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥))
10	9	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥))
11		funimassd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
12	11	3adant3 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵)
13	10, 12	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14	13	3exp 1117	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
16	6, 7, 15	rexlimd 3245	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
17	3, 16	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → 𝑦 ∈ 𝐵)
18	17	ssd 42519	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883   “ cima 5583  Fun wfun 6412  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  funimaeq  42681