Theorem resflem 5660

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 5660 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 3146	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉))
3		resflem.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉⟶𝑋)
4		fdm 5353	. . . . . . 7 ⊢ (𝐹:𝑉⟶𝑋 → dom 𝐹 = 𝑉)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝑉)
6	5	eleq2d 2240	. . . . 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 2542	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌))
12		ffun 5350	. . . 4 ⊢ (𝐹:𝑉⟶𝑋 → Fun 𝐹)
13	3, 12	syl 14	. . 3 ⊢ (𝜑 → Fun 𝐹)
14		ffvresb 5659	. . 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 1348   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121  dom cdm 4611   ↾ cres 4613  Fun wfun 5192  ⟶wf 5194  ‘cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by:  bj-charfun  13842