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Mirrors > Home > MPE Home > Th. List > nff1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
nff1.1 | ⊢ Ⅎ𝑥𝐹 |
nff1.2 | ⊢ Ⅎ𝑥𝐴 |
nff1.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nff1 | ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 5893 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
2 | nff1.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff1.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nff1.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 2, 3, 4 | nff 6041 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
6 | 2 | nfcnv 5301 | . . . 4 ⊢ Ⅎ𝑥◡𝐹 |
7 | 6 | nffun 5911 | . . 3 ⊢ Ⅎ𝑥Fun ◡𝐹 |
8 | 5, 7 | nfan 1828 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) |
9 | 1, 8 | nfxfr 1779 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 Ⅎwnf 1708 Ⅎwnfc 2751 ◡ccnv 5113 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 |
This theorem is referenced by: nff1o 6135 iundom2g 9362 |
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