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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 5189 | . 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 5385 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
6 | 2, 3, 4 | nffo 5403 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
7 | 5, 6 | nfan 1552 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) |
8 | 1, 7 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1447 Ⅎwnfc 2293 –1-1→wf1 5179 –onto→wfo 5180 –1-1-onto→wf1o 5181 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: nfiso 5768 nfsum1 11283 nfsum 11284 nfcprod1 11481 nfcprod 11482 |
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