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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 5592 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 3101 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)) |
3 | resflem.1 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉⟶𝑋) | |
4 | fdm 5286 | . . . . . . 7 ⊢ (𝐹:𝑉⟶𝑋 → dom 𝐹 = 𝑉) | |
5 | 3, 4 | syl 14 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝑉) |
6 | 5 | eleq2d 2210 | . . . . 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 2507 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑌)) |
12 | ffun 5283 | . . . 4 ⊢ (𝐹:𝑉⟶𝑋 → Fun 𝐹) | |
13 | 3, 12 | syl 14 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
14 | ffvresb 5591 | . . 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 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 dom cdm 4547 ↾ cres 4549 Fun wfun 5125 ⟶wf 5127 ‘cfv 5131 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 |
This theorem is referenced by: (None) |
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