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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 5088 | . 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 5284 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
6 | 2, 3, 4 | nffo 5302 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
7 | 5, 6 | nfan 1527 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) |
8 | 1, 7 | nfxfr 1433 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1419 Ⅎwnfc 2242 –1-1→wf1 5078 –onto→wfo 5079 –1-1-onto→wf1o 5080 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 |
This theorem is referenced by: nfiso 5661 nfsum1 11017 nfsum 11018 |
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