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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 6363 | . 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 6519 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
6 | 2 | nfcnv 5732 | . . . 4 ⊢ Ⅎ𝑥◡𝐹 |
7 | 6 | nffun 6381 | . . 3 ⊢ Ⅎ𝑥Fun ◡𝐹 |
8 | 5, 7 | nfan 1907 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) |
9 | 1, 8 | nfxfr 1860 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 Ⅎwnf 1791 Ⅎwnfc 2877 ◡ccnv 5535 Fun wfun 6352 ⟶wf 6354 –1-1→wf1 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 |
This theorem is referenced by: nff1o 6637 iundom2g 10119 |
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