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Theorem f1oiso2 6556
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 6494 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻𝑥) ∈ 𝐴)
32adantrr 752 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑥) ∈ 𝐴)
433adant3 1079 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥) ∈ 𝐴)
5 f1ocnvdm 6494 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻𝑦) ∈ 𝐴)
65adantrl 751 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑦) ∈ 𝐴)
763adant3 1079 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑦) ∈ 𝐴)
8 f1ocnvfv2 6487 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻‘(𝐻𝑥)) = 𝑥)
98eqcomd 2627 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → 𝑥 = (𝐻‘(𝐻𝑥)))
10 f1ocnvfv2 6487 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻‘(𝐻𝑦)) = 𝑦)
1110eqcomd 2627 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → 𝑦 = (𝐻‘(𝐻𝑦)))
129, 11anim12dan 881 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
13123adant3 1079 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
14 simp3 1061 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥)𝑅(𝐻𝑦))
15 fveq2 6148 . . . . . . . . . . . 12 (𝑤 = (𝐻𝑦) → (𝐻𝑤) = (𝐻‘(𝐻𝑦)))
1615eqeq2d 2631 . . . . . . . . . . 11 (𝑤 = (𝐻𝑦) → (𝑦 = (𝐻𝑤) ↔ 𝑦 = (𝐻‘(𝐻𝑦))))
1716anbi2d 739 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦)))))
18 breq2 4617 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝐻𝑥)𝑅𝑤 ↔ (𝐻𝑥)𝑅(𝐻𝑦)))
1917, 18anbi12d 746 . . . . . . . . 9 (𝑤 = (𝐻𝑦) → (((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
2019rspcev 3295 . . . . . . . 8 (((𝐻𝑦) ∈ 𝐴 ∧ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
217, 13, 14, 20syl12anc 1321 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
22 fveq2 6148 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑥) → (𝐻𝑧) = (𝐻‘(𝐻𝑥)))
2322eqeq2d 2631 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑥 = (𝐻𝑧) ↔ 𝑥 = (𝐻‘(𝐻𝑥))))
2423anbi1d 740 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤))))
25 breq1 4616 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝑅𝑤 ↔ (𝐻𝑥)𝑅𝑤))
2624, 25anbi12d 746 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2726rexbidv 3045 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (∃𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2827rspcev 3295 . . . . . . 7 (((𝐻𝑥) ∈ 𝐴 ∧ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
294, 21, 28syl2anc 692 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
30293expib 1265 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
31 simp3ll 1130 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻𝑧))
32 simp1 1059 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝐻:𝐴1-1-onto𝐵)
33 simp2l 1085 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝐴)
34 f1of 6094 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3534ffvelrnda 6315 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → (𝐻𝑧) ∈ 𝐵)
3632, 33, 35syl2anc 692 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) ∈ 𝐵)
3731, 36eqeltrd 2698 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥𝐵)
38 simp3lr 1131 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻𝑤))
39 simp2r 1086 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤𝐴)
4034ffvelrnda 6315 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → (𝐻𝑤) ∈ 𝐵)
4132, 39, 40syl2anc 692 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) ∈ 𝐵)
4238, 41eqeltrd 2698 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦𝐵)
43 simp3r 1088 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
4431eqcomd 2627 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) = 𝑥)
45 f1ocnvfv 6488 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → ((𝐻𝑧) = 𝑥 → (𝐻𝑥) = 𝑧))
4632, 33, 45syl2anc 692 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻𝑧) = 𝑥 → (𝐻𝑥) = 𝑧))
4744, 46mpd 15 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑥) = 𝑧)
4838eqcomd 2627 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) = 𝑦)
49 f1ocnvfv 6488 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → ((𝐻𝑤) = 𝑦 → (𝐻𝑦) = 𝑤))
5032, 39, 49syl2anc 692 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝐻𝑤) = 𝑦 → (𝐻𝑦) = 𝑤))
5148, 50mpd 15 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑦) = 𝑤)
5243, 47, 513brtr4d 4645 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑥)𝑅(𝐻𝑦))
5337, 42, 52jca31 556 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))
54533exp 1261 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑧𝐴𝑤𝐴) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))))
5554rexlimdvv 3030 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
5630, 55impbid 202 . . . 4 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) ↔ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
5756opabbidv 4678 . . 3 (𝐻:𝐴1-1-onto𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
581, 57syl5eq 2667 . 2 (𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
59 f1oiso 6555 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6058, 59mpdan 701 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908   class class class wbr 4613  {copab 4672  ccnv 5073  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856
This theorem is referenced by:  fnwelem  7237
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