Theorem funimassd 6957

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	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
2		fvelima 6956	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
3	1, 2	sylan 578	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
4		funimassd.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴)
6	4, 5	nfan 1900	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴))
7		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
8		id 22	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦)
9	8	eqcomd 2736	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥))
10	9	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥))
11		funimassd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
12	11	3adant3 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵)
13	10, 12	eqeltrd 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14	13	3exp 1117	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
15	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
16	6, 7, 15	rexlimd 3261	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
17	3, 16	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → 𝑦 ∈ 𝐵)
18	17	ex 411	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵))
19	18	ssrdv 3987	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  ∃wrex 3068   ⊆ wss 3947   “ cima 5678  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  ig1pmindeg  32947  funimaeq  44248