Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5810 | . 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 5901 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5289 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nffo.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfeq 2762 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵 |
8 | 4, 7 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) |
9 | 1, 8 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 Ⅎwnf 1699 Ⅎwnfc 2738 ran crn 5039 Fn wfn 5799 –onto→wfo 5802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-fo 5810 |
This theorem is referenced by: nff1o 6048 fompt 38374 |
Copyright terms: Public domain | W3C validator |