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Theorem nfiso 7064
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 6357 . 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 6606 . . 3 𝑥 𝐻:𝐴1-1-onto𝐵
6 nfcv 2974 . . . . . . 7 𝑥𝑦
7 nfiso.2 . . . . . . 7 𝑥𝑅
8 nfcv 2974 . . . . . . 7 𝑥𝑧
96, 7, 8nfbr 5104 . . . . . 6 𝑥 𝑦𝑅𝑧
102, 6nffv 6673 . . . . . . 7 𝑥(𝐻𝑦)
11 nfiso.3 . . . . . . 7 𝑥𝑆
122, 8nffv 6673 . . . . . . 7 𝑥(𝐻𝑧)
1310, 11, 12nfbr 5104 . . . . . 6 𝑥(𝐻𝑦)𝑆(𝐻𝑧)
149, 13nfbi 1895 . . . . 5 𝑥(𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
153, 14nfralw 3222 . . . 4 𝑥𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
163, 15nfralw 3222 . . 3 𝑥𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
175, 16nfan 1891 . 2 𝑥(𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧)))
181, 17nfxfr 1844 1 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wnf 1775  wnfc 2958  wral 3135   class class class wbr 5057  1-1-ontowf1o 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)
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