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Mirrors > Home > MPE Home > Th. List > nff1o | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.) |
Ref | Expression |
---|---|
nff1o.1 | ⊢ Ⅎ𝑥𝐹 |
nff1o.2 | ⊢ Ⅎ𝑥𝐴 |
nff1o.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nff1o | ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1o 6364 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
2 | nff1o.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff1o.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nff1o.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 2, 3, 4 | nff1 6575 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
6 | 2, 3, 4 | nffo 6591 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
7 | 5, 6 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) |
8 | 1, 7 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1784 Ⅎwnfc 2963 –1-1→wf1 6354 –onto→wfo 6355 –1-1-onto→wf1o 6356 |
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-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: nfiso 7077 nfsum1 15048 nfsumw 15049 nfsum 15050 nfcprod1 15266 nfcprod 15267 fsumiunle 30547 esumiun 31355 stoweidlem35 42327 |
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