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Mirrors > Home > ILE Home > Th. List > inresflem | GIF version |
Description: Lemma for inlresf1 6954 and inrresf1 6955. (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 5374 | . . 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 5376 | . . . 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 2488 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 |
8 | fnfvrnss 5588 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) | |
9 | 5, 7, 8 | mp2an 423 | . . 3 ⊢ ran 𝐹 ⊆ 𝐵 |
10 | df-f 5135 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
11 | 5, 9, 10 | mpbir2an 927 | . 2 ⊢ 𝐹:𝐴⟶𝐵 |
12 | f1ff1 5344 | . 2 ⊢ ((𝐹:𝐴–1-1→({𝑋} × 𝐴) ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵) | |
13 | 3, 11, 12 | mp2an 423 | 1 ⊢ 𝐹:𝐴–1-1→𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 {csn 3532 × cxp 4545 ran crn 4548 Fn wfn 5126 ⟶wf 5127 –1-1→wf1 5128 –1-1-onto→wf1o 5130 ‘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-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: inlresf1 6954 inrresf1 6955 |
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