Theorem funimassd 6943

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 6942	. . . . 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 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴)
6	4, 5	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴))
7		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
8		id 23	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦)
9	8	eqcomd 2767	. . . . . . . . 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 2861	. . . . . . 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 3270	. . . 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 3937	1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  ∃wrex 3087   ⊆ wss 3899   “ cima 5654  Fun wfun 6525  ‘cfv 6531

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539

This theorem is used by:  bdayiun  28283  ig1pmindeg  34116  esplylem  34180  esplyfv1  34183  exsslsb  34211  aks6d1c3  43141  aks6d1c2lem4  43145  aks6d1c2  43148  aks6d1c6lem2  43189  funimaeq  46201  tmachlem-agreesn  47901