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Theorem nfiso 5474
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 4939 . 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 5152 . . 3 𝑥 𝐻:𝐴1-1-onto𝐵
6 nfcv 2194 . . . . . . 7 𝑥𝑦
7 nfiso.2 . . . . . . 7 𝑥𝑅
8 nfcv 2194 . . . . . . 7 𝑥𝑧
96, 7, 8nfbr 3836 . . . . . 6 𝑥 𝑦𝑅𝑧
102, 6nffv 5213 . . . . . . 7 𝑥(𝐻𝑦)
11 nfiso.3 . . . . . . 7 𝑥𝑆
122, 8nffv 5213 . . . . . . 7 𝑥(𝐻𝑧)
1310, 11, 12nfbr 3836 . . . . . 6 𝑥(𝐻𝑦)𝑆(𝐻𝑧)
149, 13nfbi 1497 . . . . 5 𝑥(𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
153, 14nfralxy 2377 . . . 4 𝑥𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
163, 15nfralxy 2377 . . 3 𝑥𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧))
175, 16nfan 1473 . 2 𝑥(𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦𝑅𝑧 ↔ (𝐻𝑦)𝑆(𝐻𝑧)))
181, 17nfxfr 1379 1 𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wnf 1365  wnfc 2181  wral 2323   class class class wbr 3792  1-1-ontowf1o 4929  cfv 4930   Isom wiso 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-isom 4939
This theorem is referenced by: (None)
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