Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfiso | GIF version |
Description: Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nfiso.1 | ⊢ Ⅎ𝑥𝐻 |
nfiso.2 | ⊢ Ⅎ𝑥𝑅 |
nfiso.3 | ⊢ Ⅎ𝑥𝑆 |
nfiso.4 | ⊢ Ⅎ𝑥𝐴 |
nfiso.5 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfiso | ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-isom 5205 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)))) | |
2 | nfiso.1 | . . . 4 ⊢ Ⅎ𝑥𝐻 | |
3 | nfiso.4 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | nfiso.5 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 2, 3, 4 | nff1o 5438 | . . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵 |
6 | nfcv 2312 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
7 | nfiso.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
8 | nfcv 2312 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
9 | 6, 7, 8 | nfbr 4033 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧 |
10 | 2, 6 | nffv 5504 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦) |
11 | nfiso.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
12 | 2, 8 | nffv 5504 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧) |
13 | 10, 11, 12 | nfbr 4033 | . . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧) |
14 | 9, 13 | nfbi 1582 | . . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
15 | 3, 14 | nfralxy 2508 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
16 | 3, 15 | nfralxy 2508 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
17 | 5, 16 | nfan 1558 | . 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))) |
18 | 1, 17 | nfxfr 1467 | 1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 Ⅎwnf 1453 Ⅎwnfc 2299 ∀wral 2448 class class class wbr 3987 –1-1-onto→wf1o 5195 ‘cfv 5196 Isom wiso 5197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |