|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6321 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
| 2 | nfpred.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfpred.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
| 4 | 3 | nfcnv 5889 | . . . 4 ⊢ Ⅎ𝑥◡𝑅 | 
| 5 | nfpred.3 | . . . . 5 ⊢ Ⅎ𝑥𝑋 | |
| 6 | 5 | nfsn 4707 | . . . 4 ⊢ Ⅎ𝑥{𝑋} | 
| 7 | 4, 6 | nfima 6086 | . . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋}) | 
| 8 | 2, 7 | nfin 4224 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋})) | 
| 9 | 1, 8 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnfc 2890 ∩ cin 3950 {csn 4626 ◡ccnv 5684 “ cima 5688 Predcpred 6320 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 | 
| This theorem is referenced by: nffrecs 8308 nfwrecsOLD 8342 nfwsuc 35819 nfwlim 35823 | 
| Copyright terms: Public domain | W3C validator |