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Mirrors > Home > ILE Home > Th. List > nff1o | GIF version |
Description: Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
nff1o.1 | ⊢ Ⅎ𝑥𝐹 |
nff1o.2 | ⊢ Ⅎ𝑥𝐴 |
nff1o.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nff1o | ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 5130 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | nff1o.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff1o.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nff1o.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 2, 3, 4 | nff1 5326 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
6 | 2, 3, 4 | nffo 5344 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
7 | 5, 6 | nfan 1544 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) |
8 | 1, 7 | nfxfr 1450 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1436 Ⅎwnfc 2268 –1-1→wf1 5120 –onto→wfo 5121 –1-1-onto→wf1o 5122 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: nfiso 5707 nfsum1 11125 nfsum 11126 nfcprod1 11323 nfcprod 11324 |
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