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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 6363 | . 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 6454 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5826 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nffo.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfeq 2993 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 = 𝐵 |
8 | 4, 7 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) |
9 | 1, 8 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 Ⅎwnf 1784 Ⅎwnfc 2963 ran crn 5558 Fn wfn 6352 –onto→wfo 6355 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-fo 6363 |
This theorem is referenced by: nff1o 6615 fompt 41460 |
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