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Theorem isocnv2 5395
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 5390 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A1-1-ontoB)
2 f1ofn 5070 . . 3 (𝐻:A1-1-ontoB𝐻 Fn A)
31, 2syl 14 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻 Fn A)
4 isof1o 5390 . . 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 2554 . . . . . . . . . 10 x V
7 vex 2554 . . . . . . . . . 10 y V
86, 7brcnv 4461 . . . . . . . . 9 (x𝑅yy𝑅x)
98a1i 9 . . . . . . . 8 (((𝐻 Fn A x A) y A) → (x𝑅yy𝑅x))
10 funfvex 5135 . . . . . . . . . . 11 ((Fun 𝐻 x dom 𝐻) → (𝐻x) V)
1110funfni 4942 . . . . . . . . . 10 ((𝐻 Fn A x A) → (𝐻x) V)
1211adantr 261 . . . . . . . . 9 (((𝐻 Fn A x A) y A) → (𝐻x) V)
13 funfvex 5135 . . . . . . . . . . 11 ((Fun 𝐻 y dom 𝐻) → (𝐻y) V)
1413funfni 4942 . . . . . . . . . 10 ((𝐻 Fn A y A) → (𝐻y) V)
1514adantlr 446 . . . . . . . . 9 (((𝐻 Fn A x A) y A) → (𝐻y) V)
16 brcnvg 4459 . . . . . . . . 9 (((𝐻x) V (𝐻y) V) → ((𝐻x)𝑆(𝐻y) ↔ (𝐻y)𝑆(𝐻x)))
1712, 15, 16syl2anc 391 . . . . . . . 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 2316 . . . . . 6 ((𝐻 Fn A x A) → (y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ y A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
2019ralbidva 2316 . . . . 5 (𝐻 Fn A → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ x A y A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
21 ralcom 2467 . . . . 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 437 . . 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 4854 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB y A x A (y𝑅x ↔ (𝐻y)𝑆(𝐻x))))
25 df-isom 4854 . . 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 619 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 1390  wral 2300  Vcvv 2551   class class class wbr 3755  ccnv 4287   Fn wfn 4840  1-1-ontowf1o 4844  cfv 4845   Isom wiso 4846
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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854
This theorem is referenced by: (None)
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