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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 5813 | . 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 6048 | . . 3 ⊢ Ⅎ𝑥 𝐻:𝐴–1-1-onto→𝐵 |
6 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
7 | nfiso.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
8 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
9 | 6, 7, 8 | nfbr 4629 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦𝑅𝑧 |
10 | 2, 6 | nffv 6110 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑦) |
11 | nfiso.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑆 | |
12 | 2, 8 | nffv 6110 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑧) |
13 | 10, 11, 12 | nfbr 4629 | . . . . . 6 ⊢ Ⅎ𝑥(𝐻‘𝑦)𝑆(𝐻‘𝑧) |
14 | 9, 13 | nfbi 1821 | . . . . 5 ⊢ Ⅎ𝑥(𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
15 | 3, 14 | nfral 2929 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
16 | 3, 15 | nfral 2929 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧)) |
17 | 5, 16 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 ↔ (𝐻‘𝑦)𝑆(𝐻‘𝑧))) |
18 | 1, 17 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 Ⅎwnf 1699 Ⅎwnfc 2738 ∀wral 2896 class class class wbr 4583 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 |
This theorem is referenced by: (None) |
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