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Mirrors > Home > ILE Home > Th. List > inresflem | GIF version |
Description: Lemma for inlresf1 6946 and inrresf1 6947. (Contributed by BJ, 4-Jul-2022.) |
Ref | Expression |
---|---|
inresflem.1 | ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) |
inresflem.2 | ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵) |
Ref | Expression |
---|---|
inresflem | ⊢ 𝐹:𝐴–1-1→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inresflem.1 | . . 3 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) | |
2 | f1of1 5366 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) → 𝐹:𝐴–1-1→({𝑋} × 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝐹:𝐴–1-1→({𝑋} × 𝐴) |
4 | f1ofn 5368 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) → 𝐹 Fn 𝐴) | |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ 𝐹 Fn 𝐴 |
6 | inresflem.2 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵) | |
7 | 6 | rgen 2485 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 |
8 | fnfvrnss 5580 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) | |
9 | 5, 7, 8 | mp2an 422 | . . 3 ⊢ ran 𝐹 ⊆ 𝐵 |
10 | df-f 5127 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
11 | 5, 9, 10 | mpbir2an 926 | . 2 ⊢ 𝐹:𝐴⟶𝐵 |
12 | f1ff1 5336 | . 2 ⊢ ((𝐹:𝐴–1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) | |
13 | 3, 11, 12 | mp2an 422 | 1 ⊢ 𝐹:𝐴–1-1→𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 {csn 3527 × cxp 4537 ran crn 4540 Fn wfn 5118 ⟶wf 5119 –1-1→wf1 5120 –1-1-onto→wf1o 5122 ‘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-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-f1o 5130 df-fv 5131 |
This theorem is referenced by: inlresf1 6946 inrresf1 6947 |
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