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Theorem nfiso 6828
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 6133 . 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 6377 . . 3 𝑥 𝐻:𝐴1-1-onto𝐵
6 nfcv 2970 . . . . . . 7 𝑥𝑦
7 nfiso.2 . . . . . . 7 𝑥𝑅
8 nfcv 2970 . . . . . . 7 𝑥𝑧
96, 7, 8nfbr 4921 . . . . . 6 𝑥 𝑦𝑅𝑧
102, 6nffv 6444 . . . . . . 7 𝑥(𝐻𝑦)
11 nfiso.3 . . . . . . 7 𝑥𝑆
122, 8nffv 6444 . . . . . . 7 𝑥(𝐻𝑧)
1310, 11, 12nfbr 4921 . . . . . 6 𝑥(𝐻𝑦)𝑆(𝐻𝑧)
149, 13nfbi 2008 . . . . 5 𝑥(𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
153, 14nfral 3155 . . . 4 𝑥𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
163, 15nfral 3155 . . 3 𝑥𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
175, 16nfan 2004 . 2 𝑥(𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧)))
181, 17nfxfr 1954 1 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wnf 1884  wnfc 2957  wral 3118   class class class wbr 4874  1-1-ontowf1o 6123  cfv 6124   Isom wiso 6125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133
This theorem is referenced by: (None)
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