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Theorem f1oiso2 5545
Description: Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
Hypothesis
Ref Expression
f1oiso2.1 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))}
Assertion
Ref Expression
f1oiso2 (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)

Proof of Theorem f1oiso2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oiso2.1 . . 3 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))}
2 f1ocnvdm 5500 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻𝑥) ∈ 𝐴)
32adantrr 463 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑥) ∈ 𝐴)
433adant3 959 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥) ∈ 𝐴)
5 f1ocnvdm 5500 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻𝑦) ∈ 𝐴)
65adantrl 462 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑦) ∈ 𝐴)
763adant3 959 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑦) ∈ 𝐴)
8 f1ocnvfv2 5497 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻‘(𝐻𝑥)) = 𝑥)
98eqcomd 2088 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → 𝑥 = (𝐻‘(𝐻𝑥)))
10 f1ocnvfv2 5497 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻‘(𝐻𝑦)) = 𝑦)
1110eqcomd 2088 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → 𝑦 = (𝐻‘(𝐻𝑦)))
129, 11anim12dan 565 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
13123adant3 959 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
14 simp3 941 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥)𝑅(𝐻𝑦))
15 fveq2 5253 . . . . . . . . . . . 12 (𝑤 = (𝐻𝑦) → (𝐻𝑤) = (𝐻‘(𝐻𝑦)))
1615eqeq2d 2094 . . . . . . . . . . 11 (𝑤 = (𝐻𝑦) → (𝑦 = (𝐻𝑤) ↔ 𝑦 = (𝐻‘(𝐻𝑦))))
1716anbi2d 452 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦)))))
18 breq2 3815 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝐻𝑥)𝑅𝑤 ↔ (𝐻𝑥)𝑅(𝐻𝑦)))
1917, 18anbi12d 457 . . . . . . . . 9 (𝑤 = (𝐻𝑦) → (((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
2019rspcev 2712 . . . . . . . 8 (((𝐻𝑦) ∈ 𝐴 ∧ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
217, 13, 14, 20syl12anc 1168 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
22 fveq2 5253 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑥) → (𝐻𝑧) = (𝐻‘(𝐻𝑥)))
2322eqeq2d 2094 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑥 = (𝐻𝑧) ↔ 𝑥 = (𝐻‘(𝐻𝑥))))
2423anbi1d 453 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤))))
25 breq1 3814 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝑅𝑤 ↔ (𝐻𝑥)𝑅𝑤))
2624, 25anbi12d 457 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2726rexbidv 2375 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (∃𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2827rspcev 2712 . . . . . . 7 (((𝐻𝑥) ∈ 𝐴 ∧ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
294, 21, 28syl2anc 403 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
30293expib 1142 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
31 simp3ll 1010 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻𝑧))
32 simp1 939 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝐻:𝐴1-1-onto𝐵)
33 simp2l 965 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝐴)
34 f1of 5201 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3534ffvelrnda 5379 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → (𝐻𝑧) ∈ 𝐵)
3632, 33, 35syl2anc 403 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) ∈ 𝐵)
3731, 36eqeltrd 2159 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥𝐵)
38 simp3lr 1011 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻𝑤))
39 simp2r 966 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤𝐴)
4034ffvelrnda 5379 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → (𝐻𝑤) ∈ 𝐵)
4132, 39, 40syl2anc 403 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) ∈ 𝐵)
4238, 41eqeltrd 2159 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦𝐵)
43 simp3r 968 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
4431eqcomd 2088 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) = 𝑥)
45 f1ocnvfv 5498 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → ((𝐻𝑧) = 𝑥 → (𝐻𝑥) = 𝑧))
4632, 33, 45syl2anc 403 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻𝑧) = 𝑥 → (𝐻𝑥) = 𝑧))
4744, 46mpd 13 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑥) = 𝑧)
4838eqcomd 2088 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) = 𝑦)
49 f1ocnvfv 5498 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → ((𝐻𝑤) = 𝑦 → (𝐻𝑦) = 𝑤))
5032, 39, 49syl2anc 403 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻𝑤) = 𝑦 → (𝐻𝑦) = 𝑤))
5148, 50mpd 13 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑦) = 𝑤)
5243, 47, 513brtr4d 3841 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑥)𝑅(𝐻𝑦))
5337, 42, 52jca31 302 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))
54533exp 1138 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑧𝐴𝑤𝐴) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))))
5554rexlimdvv 2489 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
5630, 55impbid 127 . . . 4 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) ↔ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
5756opabbidv 3870 . . 3 (𝐻:𝐴1-1-onto𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
581, 57syl5eq 2127 . 2 (𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
59 f1oiso 5544 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6058, 59mpdan 412 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wrex 2354   class class class wbr 3811  {copab 3864  ccnv 4400  1-1-ontowf1o 4968  cfv 4969   Isom wiso 4970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-isom 4978
This theorem is referenced by: (None)
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