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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 5597 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | nfpred.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfpred.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
4 | 3 | nfcnv 5223 | . . . 4 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfpred.3 | . . . . 5 ⊢ Ⅎ𝑥𝑋 | |
6 | 5 | nfsn 4189 | . . . 4 ⊢ Ⅎ𝑥{𝑋} |
7 | 4, 6 | nfima 5393 | . . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋}) |
8 | 2, 7 | nfin 3782 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋})) |
9 | 1, 8 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 ∩ cin 3539 {csn 4125 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
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-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-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 |
This theorem is referenced by: nfwrecs 7296 nfwsuc 31008 nfwlim 31012 |
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