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Theorem isocnv2 5377
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
isocnv2 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆(A, B))

Proof of Theorem isocnv2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5372 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
2 f1ofn 5052 . . 3 (𝐻:A1-1-ontoB𝐻 Fn A)
31, 2syl 14 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻 Fn A)
4 isof1o 5372 . . 3 (𝐻 Isom 𝑅, 𝑆(A, B) → 𝐻:A1-1-ontoB)
54, 2syl 14 . 2 (𝐻 Isom 𝑅, 𝑆(A, B) → 𝐻 Fn A)
6 vex 2538 . . . . . . . . . 10 x V
7 vex 2538 . . . . . . . . . 10 y V
86, 7brcnv 4445 . . . . . . . . 9 (x𝑅yy𝑅x)
98a1i 9 . . . . . . . 8 (((𝐻 Fn A x A) y A) → (x𝑅yy𝑅x))
10 funfvex 5117 . . . . . . . . . . 11 ((Fun 𝐻 x dom 𝐻) → (𝐻x) V)
1110funfni 4925 . . . . . . . . . 10 ((𝐻 Fn A x A) → (𝐻x) V)
1211adantr 261 . . . . . . . . 9 (((𝐻 Fn A x A) y A) → (𝐻x) V)
13 funfvex 5117 . . . . . . . . . . 11 ((Fun 𝐻 y dom 𝐻) → (𝐻y) V)
1413funfni 4925 . . . . . . . . . 10 ((𝐻 Fn A y A) → (𝐻y) V)
1514adantlr 449 . . . . . . . . 9 (((𝐻 Fn A x A) y A) → (𝐻y) V)
16 brcnvg 4443 . . . . . . . . 9 (((𝐻x) V (𝐻y) V) → ((𝐻x)𝑆(𝐻y) ↔ (𝐻y)𝑆(𝐻x)))
1712, 15, 16syl2anc 393 . . . . . . . 8 (((𝐻 Fn A x A) y A) → ((𝐻x)𝑆(𝐻y) ↔ (𝐻y)𝑆(𝐻x)))
189, 17bibi12d 224 . . . . . . 7 (((𝐻 Fn A x A) y A) → ((x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
1918ralbidva 2300 . . . . . 6 ((𝐻 Fn A x A) → (y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ y A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
2019ralbidva 2300 . . . . 5 (𝐻 Fn A → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ x A y A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
21 ralcom 2451 . . . . 5 (y A x A (y𝑅x ↔ (𝐻y)𝑆(𝐻x)) ↔ x A y A (y𝑅x ↔ (𝐻y)𝑆(𝐻x)))
2220, 21syl6rbbr 188 . . . 4 (𝐻 Fn A → (y A x A (y𝑅x ↔ (𝐻y)𝑆(𝐻x)) ↔ x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
2322anbi2d 440 . . 3 (𝐻 Fn A → ((𝐻:A1-1-ontoB y A x A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))))
24 df-isom 4838 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB y A x A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
25 df-isom 4838 . . 3 (𝐻 Isom 𝑅, 𝑆(A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
2623, 24, 253bitr4g 212 . 2 (𝐻 Fn A → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆(A, B)))
273, 5, 26pm5.21nii 607 1 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆(A, B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1374  wral 2284  Vcvv 2535   class class class wbr 3738  ccnv 4271   Fn wfn 4824  1-1-ontowf1o 4828  cfv 4829   Isom wiso 4830
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-f1o 4836  df-fv 4837  df-isom 4838
This theorem is referenced by: (None)
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