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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 6356 | . 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 6447 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5819 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nffo.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfeq 2991 | . . 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 398 = wceq 1533 Ⅎwnf 1780 Ⅎwnfc 2961 ran crn 5551 Fn wfn 6345 –onto→wfo 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-fun 6352 df-fn 6353 df-fo 6356 |
This theorem is referenced by: nff1o 6608 fompt 41445 |
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