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Theorem nfiso 5783
Description: Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
nfiso.1 𝑥𝐻
nfiso.2 𝑥𝑅
nfiso.3 𝑥𝑆
nfiso.4 𝑥𝐴
nfiso.5 𝑥𝐵
Assertion
Ref Expression
nfiso 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Proof of Theorem nfiso
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 5205 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))))
2 nfiso.1 . . . 4 𝑥𝐻
3 nfiso.4 . . . 4 𝑥𝐴
4 nfiso.5 . . . 4 𝑥𝐵
52, 3, 4nff1o 5438 . . 3 𝑥 𝐻:𝐴1-1-onto𝐵
6 nfcv 2312 . . . . . . 7 𝑥𝑦
7 nfiso.2 . . . . . . 7 𝑥𝑅
8 nfcv 2312 . . . . . . 7 𝑥𝑧
96, 7, 8nfbr 4033 . . . . . 6 𝑥 𝑦𝑅𝑧
102, 6nffv 5504 . . . . . . 7 𝑥(𝐻𝑦)
11 nfiso.3 . . . . . . 7 𝑥𝑆
122, 8nffv 5504 . . . . . . 7 𝑥(𝐻𝑧)
1310, 11, 12nfbr 4033 . . . . . 6 𝑥(𝐻𝑦)𝑆(𝐻𝑧)
149, 13nfbi 1582 . . . . 5 𝑥(𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
153, 14nfralxy 2508 . . . 4 𝑥𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
163, 15nfralxy 2508 . . 3 𝑥𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
175, 16nfan 1558 . 2 𝑥(𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧)))
181, 17nfxfr 1467 1 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wnf 1453  wnfc 2299  wral 2448   class class class wbr 3987  1-1-ontowf1o 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)
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