Theorem resflem 5649

Description: A lemma to bound the range of a restriction. The conclusion would also hold with (𝑋 ∩ 𝑌) in place of 𝑌 (provided 𝑥 does not occur in 𝑋). If that stronger result is needed, it is however simpler to use the instance of resflem 5649 where (𝑋 ∩ 𝑌) is substituted for 𝑌 (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)

Hypotheses

Ref	Expression
resflem.1	⊢ (𝜑 → 𝐹:𝑉⟶𝑋)
resflem.2	⊢ (𝜑 → 𝐴 ⊆ 𝑉)
resflem.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑌)

Assertion

Ref	Expression
resflem	⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝑌)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐹   𝑥,𝑌

Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem resflem

Step	Hyp	Ref	Expression
1		resflem.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑉)
2	1	sseld 3141	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉))
3		resflem.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉⟶𝑋)
4		fdm 5343	. . . . . . 7 ⊢ (𝐹:𝑉⟶𝑋 → dom 𝐹 = 𝑉)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝑉)
6	5	eleq2d 2236	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝑉))
7	2, 6	sylibrd 168	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹))
8		resflem.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑌)
9	8	ex 114	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝑌))
10	7, 9	jcad 305	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌)))
11	10	ralrimiv 2538	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌))
12		ffun 5340	. . . 4 ⊢ (𝐹:𝑉⟶𝑋 → Fun 𝐹)
13	3, 12	syl 14	. . 3 ⊢ (𝜑 → Fun 𝐹)
14		ffvresb 5648	. . 3 ⊢ (Fun 𝐹 → ((𝐹 ↾ 𝐴):𝐴⟶𝑌 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌)))
15	13, 14	syl 14	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐴):𝐴⟶𝑌 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌)))
16	11, 15	mpbird 166	1 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝑌)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  dom cdm 4604   ↾ cres 4606  Fun wfun 5182  ⟶wf 5184  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by:  bj-charfun  13689