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Theorem isoun 28696
Description: Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
Hypotheses
Ref Expression
isoun.1 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
isoun.2 (𝜑𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))
isoun.3 ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)
isoun.4 ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)
isoun.5 ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)
isoun.6 ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)
isoun.7 (𝜑 → (𝐴𝐶) = ∅)
isoun.8 (𝜑 → (𝐵𝐷) = ∅)
Assertion
Ref Expression
isoun (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑤,𝑦,𝑧,𝐵   𝑥,𝐶,𝑦   𝑤,𝐷,𝑥,𝑦,𝑧   𝑤,𝐺,𝑥,𝑦,𝑧   𝑤,𝐻,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦   𝑤,𝑆,𝑥,𝑦,𝑧   𝜑,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑤)   𝐶(𝑧,𝑤)   𝑅(𝑧,𝑤)

Proof of Theorem isoun
StepHypRef Expression
1 isoun.1 . . . 4 (𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 isof1o 6451 . . . 4 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
31, 2syl 17 . . 3 (𝜑𝐻:𝐴1-1-onto𝐵)
4 isoun.2 . . . 4 (𝜑𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))
5 isof1o 6451 . . . 4 (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷) → 𝐺:𝐶1-1-onto𝐷)
64, 5syl 17 . . 3 (𝜑𝐺:𝐶1-1-onto𝐷)
7 isoun.7 . . 3 (𝜑 → (𝐴𝐶) = ∅)
8 isoun.8 . . 3 (𝜑 → (𝐵𝐷) = ∅)
9 f1oun 6054 . . 3 (((𝐻:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐻𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
103, 6, 7, 8, 9syl22anc 1318 . 2 (𝜑 → (𝐻𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
11 elun 3714 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
12 elun 3714 . . . . . . . 8 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
13 isorel 6454 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
141, 13sylan 486 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
15 f1ofn 6036 . . . . . . . . . . . . . . . 16 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
163, 15syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn 𝐴)
1716adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐻 Fn 𝐴)
18 f1ofn 6036 . . . . . . . . . . . . . . . 16 (𝐺:𝐶1-1-onto𝐷𝐺 Fn 𝐶)
196, 18syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 Fn 𝐶)
2019adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐺 Fn 𝐶)
217anim1i 589 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → ((𝐴𝐶) = ∅ ∧ 𝑥𝐴))
22 fvun1 6164 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑥𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2317, 20, 21, 22syl3anc 1317 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2423adantrr 748 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2516adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → 𝐻 Fn 𝐴)
2619adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → 𝐺 Fn 𝐶)
277anim1i 589 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → ((𝐴𝐶) = ∅ ∧ 𝑦𝐴))
28 fvun1 6164 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
2925, 26, 27, 28syl3anc 1317 . . . . . . . . . . . . 13 ((𝜑𝑦𝐴) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
3029adantrl 747 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
3124, 30breq12d 4590 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3214, 31bitr4d 269 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
3332anassrs 677 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
34 isoun.3 . . . . . . . . . . . 12 ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)
35343expb 1257 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → 𝑥𝑅𝑦)
36 isoun.4 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)
37363expia 1258 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝐵) → (𝑤𝐷𝑧𝑆𝑤))
3837ralrimiv 2947 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐵) → ∀𝑤𝐷 𝑧𝑆𝑤)
3938ralrimiva 2948 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤)
4039adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤)
41 f1of 6035 . . . . . . . . . . . . . . . . 17 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
423, 41syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐻:𝐴𝐵)
4342ffvelrnda 6252 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
4443adantrr 748 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐻𝑥) ∈ 𝐵)
45 f1of 6035 . . . . . . . . . . . . . . . . 17 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶𝐷)
466, 45syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐺:𝐶𝐷)
4746ffvelrnda 6252 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐶) → (𝐺𝑦) ∈ 𝐷)
4847adantrl 747 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐺𝑦) ∈ 𝐷)
49 breq1 4580 . . . . . . . . . . . . . . 15 (𝑧 = (𝐻𝑥) → (𝑧𝑆𝑤 ↔ (𝐻𝑥)𝑆𝑤))
50 breq2 4581 . . . . . . . . . . . . . . 15 (𝑤 = (𝐺𝑦) → ((𝐻𝑥)𝑆𝑤 ↔ (𝐻𝑥)𝑆(𝐺𝑦)))
5149, 50rspc2v 3292 . . . . . . . . . . . . . 14 (((𝐻𝑥) ∈ 𝐵 ∧ (𝐺𝑦) ∈ 𝐷) → (∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤 → (𝐻𝑥)𝑆(𝐺𝑦)))
5244, 48, 51syl2anc 690 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤 → (𝐻𝑥)𝑆(𝐺𝑦)))
5340, 52mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐻𝑥)𝑆(𝐺𝑦))
5423adantrr 748 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
5516adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → 𝐻 Fn 𝐴)
5619adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → 𝐺 Fn 𝐶)
577anim1i 589 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → ((𝐴𝐶) = ∅ ∧ 𝑦𝐶))
58 fvun2 6165 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
5955, 56, 57, 58syl3anc 1317 . . . . . . . . . . . . 13 ((𝜑𝑦𝐶) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
6059adantrl 747 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
6153, 54, 603brtr4d 4609 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))
6235, 612thd 253 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6362anassrs 677 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6433, 63jaodan 821 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ (𝑦𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6512, 64sylan2b 490 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6665ex 448 . . . . . 6 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
67 isoun.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)
68673expb 1257 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ 𝑥𝑅𝑦)
69 isoun.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)
70693expia 1258 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝐷) → (𝑤𝐵 → ¬ 𝑧𝑆𝑤))
7170ralrimiv 2947 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐷) → ∀𝑤𝐵 ¬ 𝑧𝑆𝑤)
7271ralrimiva 2948 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤)
7372adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤)
7446ffvelrnda 6252 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → (𝐺𝑥) ∈ 𝐷)
7574adantrr 748 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝐺𝑥) ∈ 𝐷)
7642ffvelrnda 6252 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
7776adantrl 747 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
78 breq1 4580 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑥) → (𝑧𝑆𝑤 ↔ (𝐺𝑥)𝑆𝑤))
7978notbid 306 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑥) → (¬ 𝑧𝑆𝑤 ↔ ¬ (𝐺𝑥)𝑆𝑤))
80 breq2 4581 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐻𝑦) → ((𝐺𝑥)𝑆𝑤 ↔ (𝐺𝑥)𝑆(𝐻𝑦)))
8180notbid 306 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑦) → (¬ (𝐺𝑥)𝑆𝑤 ↔ ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8279, 81rspc2v 3292 . . . . . . . . . . . . . 14 (((𝐺𝑥) ∈ 𝐷 ∧ (𝐻𝑦) ∈ 𝐵) → (∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8375, 77, 82syl2anc 690 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8473, 83mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ (𝐺𝑥)𝑆(𝐻𝑦))
8516adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → 𝐻 Fn 𝐴)
8619adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → 𝐺 Fn 𝐶)
877anim1i 589 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → ((𝐴𝐶) = ∅ ∧ 𝑥𝐶))
88 fvun2 6165 . . . . . . . . . . . . . . 15 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑥𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
8985, 86, 87, 88syl3anc 1317 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐶) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9089adantrr 748 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9129adantrl 747 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
9290, 91breq12d 4590 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐺𝑥)𝑆(𝐻𝑦)))
9384, 92mtbird 313 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))
9468, 932falsed 364 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
9594anassrs 677 . . . . . . . . 9 (((𝜑𝑥𝐶) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
96 isorel 6454 . . . . . . . . . . . 12 ((𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
974, 96sylan 486 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
9889adantrr 748 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9959adantrl 747 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
10098, 99breq12d 4590 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
10197, 100bitr4d 269 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
102101anassrs 677 . . . . . . . . 9 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
10395, 102jaodan 821 . . . . . . . 8 (((𝜑𝑥𝐶) ∧ (𝑦𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
10412, 103sylan2b 490 . . . . . . 7 (((𝜑𝑥𝐶) ∧ 𝑦 ∈ (𝐴𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
105104ex 448 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
10666, 105jaodan 821 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐶)) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
10711, 106sylan2b 490 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
108107ralrimiv 2947 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → ∀𝑦 ∈ (𝐴𝐶)(𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
109108ralrimiva 2948 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)∀𝑦 ∈ (𝐴𝐶)(𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
110 df-isom 5799 . 2 ((𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)) ↔ ((𝐻𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷) ∧ ∀𝑥 ∈ (𝐴𝐶)∀𝑦 ∈ (𝐴𝐶)(𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
11110, 109, 110sylanbrc 694 1 (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cun 3537  cin 3538  c0 3873   class class class wbr 4577   Fn wfn 5785  wf 5786  1-1-ontowf1o 5789  cfv 5790   Isom wiso 5791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799
This theorem is referenced by: (None)
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