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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 6360 | . 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 6510 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
6 | 2 | nfcnv 5749 | . . . 4 ⊢ Ⅎ𝑥◡𝐹 |
7 | 6 | nffun 6378 | . . 3 ⊢ Ⅎ𝑥Fun ◡𝐹 |
8 | 5, 7 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) |
9 | 1, 8 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1784 Ⅎwnfc 2961 ◡ccnv 5554 Fun wfun 6349 ⟶wf 6351 –1-1→wf1 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 |
This theorem is referenced by: nff1o 6613 iundom2g 9962 |
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