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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 5660 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 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) |
Colors of variables: wff set class |
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 |
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