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Theorem nfiso 7268
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 6506 . 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 6783 . . 3 𝑥 𝐻:𝐴1-1-onto𝐵
6 nfcv 2904 . . . . . . 7 𝑥𝑦
7 nfiso.2 . . . . . . 7 𝑥𝑅
8 nfcv 2904 . . . . . . 7 𝑥𝑧
96, 7, 8nfbr 5153 . . . . . 6 𝑥 𝑦𝑅𝑧
102, 6nffv 6853 . . . . . . 7 𝑥(𝐻𝑦)
11 nfiso.3 . . . . . . 7 𝑥𝑆
122, 8nffv 6853 . . . . . . 7 𝑥(𝐻𝑧)
1310, 11, 12nfbr 5153 . . . . . 6 𝑥(𝐻𝑦)𝑆(𝐻𝑧)
149, 13nfbi 1907 . . . . 5 𝑥(𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
153, 14nfralw 3293 . . . 4 𝑥𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
163, 15nfralw 3293 . . 3 𝑥𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
175, 16nfan 1903 . 2 𝑥(𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧)))
181, 17nfxfr 1856 1 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wnf 1786  wnfc 2884  wral 3061   class class class wbr 5106  1-1-ontowf1o 6496  cfv 6497   Isom wiso 6498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506
This theorem is referenced by: (None)
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