Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfiso | Structured version Visualization version 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 6357 | . 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 6606 | . . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵 |
6 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
7 | nfiso.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
8 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
9 | 6, 7, 8 | nfbr 5104 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧 |
10 | 2, 6 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦) |
11 | nfiso.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
12 | 2, 8 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧) |
13 | 10, 11, 12 | nfbr 5104 | . . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧) |
14 | 9, 13 | nfbi 1895 | . . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
15 | 3, 14 | nfralw 3222 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
16 | 3, 15 | nfralw 3222 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
17 | 5, 16 | nfan 1891 | . 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))) |
18 | 1, 17 | nfxfr 1844 | 1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 Ⅎwnf 1775 Ⅎwnfc 2958 ∀wral 3135 class class class wbr 5057 –1-1-onto→wf1o 6347 ‘cfv 6348 Isom wiso 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |