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Theorem isoun 30461
 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 7055 . . . 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 7055 . . . 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 6609 . . 3 (((𝐻:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐻𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
103, 6, 7, 8, 9syl22anc 837 . 2 (𝜑 → (𝐻𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
11 elun 4076 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
12 elun 4076 . . . . . . . 8 (𝑦 ∈ (𝐴𝐶) ↔ (𝑦𝐴𝑦𝐶))
13 isorel 7058 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
141, 13sylan 583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
15 f1ofn 6591 . . . . . . . . . . . . . . . 16 (𝐻:𝐴1-1-onto𝐵𝐻 Fn 𝐴)
163, 15syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐻 Fn 𝐴)
1716adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐻 Fn 𝐴)
18 f1ofn 6591 . . . . . . . . . . . . . . . 16 (𝐺:𝐶1-1-onto𝐷𝐺 Fn 𝐶)
196, 18syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐺 Fn 𝐶)
2019adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐺 Fn 𝐶)
217anim1i 617 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → ((𝐴𝐶) = ∅ ∧ 𝑥𝐴))
22 fvun1 6729 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑥𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2317, 20, 21, 22syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2423adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
2516adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → 𝐻 Fn 𝐴)
2619adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → 𝐺 Fn 𝐶)
277anim1i 617 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → ((𝐴𝐶) = ∅ ∧ 𝑦𝐴))
28 fvun1 6729 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
2925, 26, 27, 28syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑𝑦𝐴) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
3029adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
3124, 30breq12d 5043 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3214, 31bitr4d 285 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
3332anassrs 471 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
34 isoun.3 . . . . . . . . . . . 12 ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)
35343expb 1117 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → 𝑥𝑅𝑦)
36 isoun.4 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)
37363expia 1118 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝐵) → (𝑤𝐷𝑧𝑆𝑤))
3837ralrimiv 3148 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐵) → ∀𝑤𝐷 𝑧𝑆𝑤)
3938ralrimiva 3149 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤)
4039adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤)
41 f1of 6590 . . . . . . . . . . . . . . . . 17 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
423, 41syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐻:𝐴𝐵)
4342ffvelrnda 6828 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → (𝐻𝑥) ∈ 𝐵)
4443adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐻𝑥) ∈ 𝐵)
45 f1of 6590 . . . . . . . . . . . . . . . . 17 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶𝐷)
466, 45syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐺:𝐶𝐷)
4746ffvelrnda 6828 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐶) → (𝐺𝑦) ∈ 𝐷)
4847adantrl 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐺𝑦) ∈ 𝐷)
49 breq1 5033 . . . . . . . . . . . . . . 15 (𝑧 = (𝐻𝑥) → (𝑧𝑆𝑤 ↔ (𝐻𝑥)𝑆𝑤))
50 breq2 5034 . . . . . . . . . . . . . . 15 (𝑤 = (𝐺𝑦) → ((𝐻𝑥)𝑆𝑤 ↔ (𝐻𝑥)𝑆(𝐺𝑦)))
5149, 50rspc2v 3581 . . . . . . . . . . . . . 14 (((𝐻𝑥) ∈ 𝐵 ∧ (𝐺𝑦) ∈ 𝐷) → (∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤 → (𝐻𝑥)𝑆(𝐺𝑦)))
5244, 48, 51syl2anc 587 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (∀𝑧𝐵𝑤𝐷 𝑧𝑆𝑤 → (𝐻𝑥)𝑆(𝐺𝑦)))
5340, 52mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝐻𝑥)𝑆(𝐺𝑦))
5423adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐻𝑥))
5516adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → 𝐻 Fn 𝐴)
5619adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → 𝐺 Fn 𝐶)
577anim1i 617 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐶) → ((𝐴𝐶) = ∅ ∧ 𝑦𝐶))
58 fvun2 6730 . . . . . . . . . . . . . 14 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
5955, 56, 57, 58syl3anc 1368 . . . . . . . . . . . . 13 ((𝜑𝑦𝐶) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
6059adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
6153, 54, 603brtr4d 5062 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))
6235, 612thd 268 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6362anassrs 471 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6433, 63jaodan 955 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ (𝑦𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6512, 64sylan2b 596 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
6665ex 416 . . . . . 6 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
67 isoun.5 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)
68673expb 1117 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ 𝑥𝑅𝑦)
69 isoun.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)
70693expia 1118 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝐷) → (𝑤𝐵 → ¬ 𝑧𝑆𝑤))
7170ralrimiv 3148 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐷) → ∀𝑤𝐵 ¬ 𝑧𝑆𝑤)
7271ralrimiva 3149 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤)
7372adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤)
7446ffvelrnda 6828 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → (𝐺𝑥) ∈ 𝐷)
7574adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝐺𝑥) ∈ 𝐷)
7642ffvelrnda 6828 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → (𝐻𝑦) ∈ 𝐵)
7776adantrl 715 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝐻𝑦) ∈ 𝐵)
78 breq1 5033 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐺𝑥) → (𝑧𝑆𝑤 ↔ (𝐺𝑥)𝑆𝑤))
7978notbid 321 . . . . . . . . . . . . . . 15 (𝑧 = (𝐺𝑥) → (¬ 𝑧𝑆𝑤 ↔ ¬ (𝐺𝑥)𝑆𝑤))
80 breq2 5034 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐻𝑦) → ((𝐺𝑥)𝑆𝑤 ↔ (𝐺𝑥)𝑆(𝐻𝑦)))
8180notbid 321 . . . . . . . . . . . . . . 15 (𝑤 = (𝐻𝑦) → (¬ (𝐺𝑥)𝑆𝑤 ↔ ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8279, 81rspc2v 3581 . . . . . . . . . . . . . 14 (((𝐺𝑥) ∈ 𝐷 ∧ (𝐻𝑦) ∈ 𝐵) → (∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8375, 77, 82syl2anc 587 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (∀𝑧𝐷𝑤𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺𝑥)𝑆(𝐻𝑦)))
8473, 83mpd 15 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ (𝐺𝑥)𝑆(𝐻𝑦))
8516adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → 𝐻 Fn 𝐴)
8619adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → 𝐺 Fn 𝐶)
877anim1i 617 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐶) → ((𝐴𝐶) = ∅ ∧ 𝑥𝐶))
88 fvun2 6730 . . . . . . . . . . . . . . 15 ((𝐻 Fn 𝐴𝐺 Fn 𝐶 ∧ ((𝐴𝐶) = ∅ ∧ 𝑥𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
8985, 86, 87, 88syl3anc 1368 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐶) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9089adantrr 716 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9129adantrl 715 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ((𝐻𝐺)‘𝑦) = (𝐻𝑦))
9290, 91breq12d 5043 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐺𝑥)𝑆(𝐻𝑦)))
9384, 92mtbird 328 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → ¬ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))
9468, 932falsed 380 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
9594anassrs 471 . . . . . . . . 9 (((𝜑𝑥𝐶) ∧ 𝑦𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
96 isorel 7058 . . . . . . . . . . . 12 ((𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
974, 96sylan 583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
9889adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐻𝐺)‘𝑥) = (𝐺𝑥))
9959adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝐻𝐺)‘𝑦) = (𝐺𝑦))
10098, 99breq12d 5043 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦) ↔ (𝐺𝑥)𝑆(𝐺𝑦)))
10197, 100bitr4d 285 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
102101anassrs 471 . . . . . . . . 9 (((𝜑𝑥𝐶) ∧ 𝑦𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
10395, 102jaodan 955 . . . . . . . 8 (((𝜑𝑥𝐶) ∧ (𝑦𝐴𝑦𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
10412, 103sylan2b 596 . . . . . . 7 (((𝜑𝑥𝐶) ∧ 𝑦 ∈ (𝐴𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
105104ex 416 . . . . . 6 ((𝜑𝑥𝐶) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
10666, 105jaodan 955 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑥𝐶)) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
10711, 106sylan2b 596 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑦 ∈ (𝐴𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦))))
108107ralrimiv 3148 . . 3 ((𝜑𝑥 ∈ (𝐴𝐶)) → ∀𝑦 ∈ (𝐴𝐶)(𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
109108ralrimiva 3149 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)∀𝑦 ∈ (𝐴𝐶)(𝑥𝑅𝑦 ↔ ((𝐻𝐺)‘𝑥)𝑆((𝐻𝐺)‘𝑦)))
110 df-isom 6333 . 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 586 1 (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∪ cun 3879   ∩ cin 3880  ∅c0 4243   class class class wbr 5030   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324   Isom wiso 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333 This theorem is referenced by: (None)
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