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Theorem nfiso 5387
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 4853 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB y A z A (y𝑅z ↔ (𝐻y)𝑆(𝐻z))))
2 nfiso.1 . . . 4 x𝐻
3 nfiso.4 . . . 4 xA
4 nfiso.5 . . . 4 xB
52, 3, 4nff1o 5065 . . 3 x 𝐻:A1-1-ontoB
6 nfcv 2175 . . . . . . 7 xy
7 nfiso.2 . . . . . . 7 x𝑅
8 nfcv 2175 . . . . . . 7 xz
96, 7, 8nfbr 3798 . . . . . 6 x y𝑅z
102, 6nffv 5126 . . . . . . 7 x(𝐻y)
11 nfiso.3 . . . . . . 7 x𝑆
122, 8nffv 5126 . . . . . . 7 x(𝐻z)
1310, 11, 12nfbr 3798 . . . . . 6 x(𝐻y)𝑆(𝐻z)
149, 13nfbi 1478 . . . . 5 x(y𝑅z ↔ (𝐻y)𝑆(𝐻z))
153, 14nfralxy 2354 . . . 4 xz A (y𝑅z ↔ (𝐻y)𝑆(𝐻z))
163, 15nfralxy 2354 . . 3 xy A z A (y𝑅z ↔ (𝐻y)𝑆(𝐻z))
175, 16nfan 1454 . 2 x(𝐻:A1-1-ontoB y A z A (y𝑅z ↔ (𝐻y)𝑆(𝐻z)))
181, 17nfxfr 1360 1 x 𝐻 Isom 𝑅, 𝑆 (A, B)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wnf 1346  wnfc 2162  wral 2300   class class class wbr 3754  1-1-ontowf1o 4843  cfv 4844   Isom wiso 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-isom 4853
This theorem is referenced by: (None)
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