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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 5811 | . 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 6012 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
6 | 2, 3, 4 | nffo 6027 | . . 3 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
7 | 5, 6 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵) |
8 | 1, 7 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 Ⅎwnf 1699 Ⅎwnfc 2738 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 |
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-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: nfiso 6472 nfsum1 14268 nfsum 14269 nfcprod1 14479 nfcprod 14480 esumiun 29483 wessf1ornlem 38366 stoweidlem35 38928 |
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