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Mirrors > Home > MPE Home > Th. List > nfpred | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.) |
Ref | Expression |
---|---|
nfpred.1 | ⊢ Ⅎ𝑥𝑅 |
nfpred.2 | ⊢ Ⅎ𝑥𝐴 |
nfpred.3 | ⊢ Ⅎ𝑥𝑋 |
Ref | Expression |
---|---|
nfpred | ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6150 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | nfpred.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfpred.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
4 | 3 | nfcnv 5751 | . . . 4 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfpred.3 | . . . . 5 ⊢ Ⅎ𝑥𝑋 | |
6 | 5 | nfsn 4645 | . . . 4 ⊢ Ⅎ𝑥{𝑋} |
7 | 4, 6 | nfima 5939 | . . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋}) |
8 | 2, 7 | nfin 4195 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋})) |
9 | 1, 8 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2963 ∩ cin 3937 {csn 4569 ◡ccnv 5556 “ cima 5560 Predcpred 6149 |
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-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-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 |
This theorem is referenced by: nfwrecs 7951 nfwsuc 33107 nfwlim 33111 nffrecs 33122 |
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