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Mirrors > Home > MPE Home > Th. List > nffun | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
nffun.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nffun | ⊢ Ⅎ𝑥Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 6542 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | nffun.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfrel 5777 | . . 3 ⊢ Ⅎ𝑥Rel 𝐹 |
4 | 2 | nfcnv 5876 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
5 | 2, 4 | nfco 5863 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) |
6 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥 I | |
7 | 5, 6 | nfss 3973 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) ⊆ I |
8 | 3, 7 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) |
9 | 1, 8 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 Ⅎwnf 1786 Ⅎwnfc 2884 ⊆ wss 3947 I cid 5572 ◡ccnv 5674 ∘ ccom 5679 Rel wrel 5680 Fun wfun 6534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-rel 5682 df-cnv 5683 df-co 5684 df-fun 6542 |
This theorem is referenced by: nffn 6645 nff1 6782 fliftfun 7304 funimass4f 31839 nfdfat 45770 |
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