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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 6015 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | nfpred.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfpred.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
4 | 3 | nfcnv 5627 | . . . 4 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfpred.3 | . . . . 5 ⊢ Ⅎ𝑥𝑋 | |
6 | 5 | nfsn 4544 | . . . 4 ⊢ Ⅎ𝑥{𝑋} |
7 | 4, 6 | nfima 5806 | . . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋}) |
8 | 2, 7 | nfin 4108 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋})) |
9 | 1, 8 | nfcxfr 2945 | 1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2931 ∩ cin 3853 {csn 4466 ◡ccnv 5434 “ cima 5438 Predcpred 6014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-opab 5019 df-xp 5441 df-cnv 5443 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 |
This theorem is referenced by: nfwrecs 7791 nfwsuc 32659 nfwlim 32663 nffrecs 32674 |
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