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Mirrors > Home > ILE Home > Th. List > resflem | GIF version |
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 5584 where (𝑋 ∩ 𝑌) is substituted for 𝑌 (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.) |
Ref | Expression |
---|---|
resflem.1 | ⊢ (𝜑 → 𝐹:𝑉⟶𝑋) |
resflem.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝑉) |
resflem.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝑌) |
Ref | Expression |
---|---|
resflem | ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resflem.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑉) | |
2 | 1 | sseld 3096 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)) |
3 | resflem.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉⟶𝑋) | |
4 | fdm 5278 | . . . . . . 7 ⊢ (𝐹:𝑉⟶𝑋 → dom 𝐹 = 𝑉) | |
5 | 3, 4 | syl 14 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝑉) |
6 | 5 | eleq2d 2209 | . . . . 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 2504 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌)) |
12 | ffun 5275 | . . . 4 ⊢ (𝐹:𝑉⟶𝑋 → Fun 𝐹) | |
13 | 3, 12 | syl 14 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
14 | ffvresb 5583 | . . 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) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 dom cdm 4539 ↾ cres 4541 Fun wfun 5117 ⟶wf 5119 ‘cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: (None) |
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