![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fneq1d | GIF version |
Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
fneq1d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
fneq1d | ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1d.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | fneq1 5304 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 Fn wfn 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-fun 5218 df-fn 5219 |
This theorem is referenced by: fneq12d 5308 f1o00 5496 f1ompt 5667 fmpt2d 5678 f1ocnvd 6072 offval2 6097 ofrfval2 6098 caofinvl 6104 f1od2 6235 cc3 7266 plusffng 12778 grpinvfng 12911 grpinvf1o 12934 mulgfng 12981 srg1zr 13163 neif 13572 fnmptd 14476 |
Copyright terms: Public domain | W3C validator |