Theorem funimassd 6977

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 6976	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
3	1, 2	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
4		funimassd.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
5		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴)
6	4, 5	nfan 1934	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴))
7		nfv 1949	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
8		id 23	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦)
9	8	eqcomd 2769	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥))
10	9	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥))
11		funimassd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
12	11	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵)
13	10, 12	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵)
14	13	3exp 1137	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
15	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)))
16	6, 7, 15	rexlimd 3273	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))
17	3, 16	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → 𝑦 ∈ 𝐵)
18	17	ex 418	. 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵))
19	18	ssrdv 3919	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   “ cima 5659  Fun wfun 6555  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569

This theorem is used by:  bdayiun  28334  ig1pmindeg  34217  esplylem  34281  esplyfv1  34284  exsslsb  34312  aks6d1c3  43376  aks6d1c2lem4  43380  aks6d1c2  43383  aks6d1c6lem2  43424  funimaeq  46473  tmachlem-agreesn  48172