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Theorem f1oiso2 7161
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 7095 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻𝑥) ∈ 𝐴)
32adantrr 717 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑥) ∈ 𝐴)
433adant3 1134 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥) ∈ 𝐴)
5 f1ocnvdm 7095 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻𝑦) ∈ 𝐴)
65adantrl 716 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝐻𝑦) ∈ 𝐴)
763adant3 1134 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑦) ∈ 𝐴)
8 f1ocnvfv2 7088 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → (𝐻‘(𝐻𝑥)) = 𝑥)
98eqcomd 2743 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑥𝐵) → 𝑥 = (𝐻‘(𝐻𝑥)))
10 f1ocnvfv2 7088 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → (𝐻‘(𝐻𝑦)) = 𝑦)
1110eqcomd 2743 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑦𝐵) → 𝑦 = (𝐻‘(𝐻𝑦)))
129, 11anim12dan 622 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
13123adant3 1134 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))))
14 simp3 1140 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → (𝐻𝑥)𝑅(𝐻𝑦))
15 fveq2 6717 . . . . . . . . . . . 12 (𝑤 = (𝐻𝑦) → (𝐻𝑤) = (𝐻‘(𝐻𝑦)))
1615eqeq2d 2748 . . . . . . . . . . 11 (𝑤 = (𝐻𝑦) → (𝑦 = (𝐻𝑤) ↔ 𝑦 = (𝐻‘(𝐻𝑦))))
1716anbi2d 632 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦)))))
18 breq2 5057 . . . . . . . . . 10 (𝑤 = (𝐻𝑦) → ((𝐻𝑥)𝑅𝑤 ↔ (𝐻𝑥)𝑅(𝐻𝑦)))
1917, 18anbi12d 634 . . . . . . . . 9 (𝑤 = (𝐻𝑦) → (((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
2019rspcev 3537 . . . . . . . 8 (((𝐻𝑦) ∈ 𝐴 ∧ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻‘(𝐻𝑦))) ∧ (𝐻𝑥)𝑅(𝐻𝑦))) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
217, 13, 14, 20syl12anc 837 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤))
22 fveq2 6717 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑥) → (𝐻𝑧) = (𝐻‘(𝐻𝑥)))
2322eqeq2d 2748 . . . . . . . . . . 11 (𝑧 = (𝐻𝑥) → (𝑥 = (𝐻𝑧) ↔ 𝑥 = (𝐻‘(𝐻𝑥))))
2423anbi1d 633 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ↔ (𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤))))
25 breq1 5056 . . . . . . . . . 10 (𝑧 = (𝐻𝑥) → (𝑧𝑅𝑤 ↔ (𝐻𝑥)𝑅𝑤))
2624, 25anbi12d 634 . . . . . . . . 9 (𝑧 = (𝐻𝑥) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2726rexbidv 3216 . . . . . . . 8 (𝑧 = (𝐻𝑥) → (∃𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) ↔ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)))
2827rspcev 3537 . . . . . . 7 (((𝐻𝑥) ∈ 𝐴 ∧ ∃𝑤𝐴 ((𝑥 = (𝐻‘(𝐻𝑥)) ∧ 𝑦 = (𝐻𝑤)) ∧ (𝐻𝑥)𝑅𝑤)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
294, 21, 28syl2anc 587 . . . . . 6 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤))
30293expib 1124 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) → ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
31 simp3ll 1246 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥 = (𝐻𝑧))
32 simp1 1138 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝐻:𝐴1-1-onto𝐵)
33 simp2l 1201 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝐴)
34 f1of 6661 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
3534ffvelrnda 6904 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → (𝐻𝑧) ∈ 𝐵)
3632, 33, 35syl2anc 587 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) ∈ 𝐵)
3731, 36eqeltrd 2838 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑥𝐵)
38 simp3lr 1247 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦 = (𝐻𝑤))
39 simp2r 1202 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑤𝐴)
4034ffvelrnda 6904 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → (𝐻𝑤) ∈ 𝐵)
4132, 39, 40syl2anc 587 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) ∈ 𝐵)
4238, 41eqeltrd 2838 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑦𝐵)
43 simp3r 1204 . . . . . . . . 9 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → 𝑧𝑅𝑤)
4431eqcomd 2743 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑧) = 𝑥)
45 f1ocnvfv 7089 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑧𝐴) → ((𝐻𝑧) = 𝑥 → (𝐻𝑥) = 𝑧))
4632, 33, 45syl2anc 587 . . . . . . . . . 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 2743 . . . . . . . . . 10 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑤) = 𝑦)
49 f1ocnvfv 7089 . . . . . . . . . . 11 ((𝐻:𝐴1-1-onto𝐵𝑤𝐴) → ((𝐻𝑤) = 𝑦 → (𝐻𝑦) = 𝑤))
5032, 39, 49syl2anc 587 . . . . . . . . . 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 5085 . . . . . . . 8 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → (𝐻𝑥)𝑅(𝐻𝑦))
5337, 42, 52jca31 518 . . . . . . 7 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑧𝐴𝑤𝐴) ∧ ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))
54533exp 1121 . . . . . 6 (𝐻:𝐴1-1-onto𝐵 → ((𝑧𝐴𝑤𝐴) → (((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)))))
5554rexlimdvv 3212 . . . . 5 (𝐻:𝐴1-1-onto𝐵 → (∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤) → ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))))
5630, 55impbid 215 . . . 4 (𝐻:𝐴1-1-onto𝐵 → (((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦)) ↔ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)))
5756opabbidv 5119 . . 3 (𝐻:𝐴1-1-onto𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
581, 57eqtrid 2789 . 2 (𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)})
59 f1oiso 7160 . 2 ((𝐻:𝐴1-1-onto𝐵𝑆 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴𝑤𝐴 ((𝑥 = (𝐻𝑧) ∧ 𝑦 = (𝐻𝑤)) ∧ 𝑧𝑅𝑤)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
6058, 59mpdan 687 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wrex 3062   class class class wbr 5053  {copab 5115  ccnv 5550  1-1-ontowf1o 6379  cfv 6380   Isom wiso 6381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389
This theorem is referenced by:  fnwelem  7898
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