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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 6133 | . 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 6377 | . . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵 |
6 | nfcv 2970 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
7 | nfiso.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
8 | nfcv 2970 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
9 | 6, 7, 8 | nfbr 4921 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧 |
10 | 2, 6 | nffv 6444 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦) |
11 | nfiso.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
12 | 2, 8 | nffv 6444 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧) |
13 | 10, 11, 12 | nfbr 4921 | . . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧) |
14 | 9, 13 | nfbi 2008 | . . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
15 | 3, 14 | nfral 3155 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
16 | 3, 15 | nfral 3155 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
17 | 5, 16 | nfan 2004 | . 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))) |
18 | 1, 17 | nfxfr 1954 | 1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 Ⅎwnf 1884 Ⅎwnfc 2957 ∀wral 3118 class class class wbr 4874 –1-1-onto→wf1o 6123 ‘cfv 6124 Isom wiso 6125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 |
This theorem is referenced by: (None) |
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