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Mirrors > Home > MPE Home > Th. List > nffo | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
nffo.1 | ⊢ Ⅎ𝑥𝐹 |
nffo.2 | ⊢ Ⅎ𝑥𝐴 |
nffo.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nffo | ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fo 6568 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
2 | nffo.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nffo.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffn 6667 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5965 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nffo.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfeq 2916 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵 |
8 | 4, 7 | nfan 1896 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) |
9 | 1, 8 | nfxfr 1849 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 Ⅎwnf 1779 Ⅎwnfc 2887 ran crn 5689 Fn wfn 6557 –onto→wfo 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-fo 6568 |
This theorem is referenced by: nff1o 6846 fompt 7137 |
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