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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 6364 | . 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 6613 | . . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵 |
6 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
7 | nfiso.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
8 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
9 | 6, 7, 8 | nfbr 5113 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧 |
10 | 2, 6 | nffv 6680 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦) |
11 | nfiso.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
12 | 2, 8 | nffv 6680 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧) |
13 | 10, 11, 12 | nfbr 5113 | . . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧) |
14 | 9, 13 | nfbi 1904 | . . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
15 | 3, 14 | nfralw 3225 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
16 | 3, 15 | nfralw 3225 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
17 | 5, 16 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))) |
18 | 1, 17 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 Ⅎwnf 1784 Ⅎwnfc 2961 ∀wral 3138 class class class wbr 5066 –1-1-onto→wf1o 6354 ‘cfv 6355 Isom wiso 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 |
This theorem is referenced by: (None) |
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