Theorem isoun 32678

Description: Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)

Hypotheses

Ref	Expression
isoun.1	⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
isoun.2	⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))
isoun.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦)
isoun.4	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤)
isoun.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦)
isoun.6	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤)
isoun.7	⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
isoun.8	⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)

Assertion

Ref	Expression
isoun	⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷)))

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

Step	Hyp	Ref	Expression
1		isoun.1	. . . 4 ⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		isof1o 7257	. . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐵)
4		isoun.2	. . . 4 ⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))
5		isof1o 7257	. . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷) → 𝐺:𝐶–1-1-onto→𝐷)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝐶–1-1-onto→𝐷)
7		isoun.7	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
8		isoun.8	. . 3 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)
9		f1oun 6782	. . 3 ⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐻 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))
10	3, 6, 7, 8, 9	syl22anc 838	. 2 ⊢ (𝜑 → (𝐻 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))
11		elun 4103	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐶) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶))
12		elun 4103	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐶) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶))
13		isorel 7260	. . . . . . . . . . . 12 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
14	1, 13	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
15		f1ofn 6764	. . . . . . . . . . . . . . . 16 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Fn 𝐴)
16	3, 15	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 Fn 𝐴)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 Fn 𝐴)
18		f1ofn 6764	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺 Fn 𝐶)
19	6, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 Fn 𝐶)
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺 Fn 𝐶)
21	7	anim1i 615	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑥 ∈ 𝐴))
22		fvun1 6913	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐺 Fn 𝐶 ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑥 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐻‘𝑥))
23	17, 20, 21, 22	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐻‘𝑥))
24	23	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐻‘𝑥))
25	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐻 Fn 𝐴)
26	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐺 Fn 𝐶)
27	7	anim1i 615	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑦 ∈ 𝐴))
28		fvun1 6913	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐺 Fn 𝐶 ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑦 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐻‘𝑦))
29	25, 26, 27, 28	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐻‘𝑦))
30	29	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐻‘𝑦))
31	24, 30	breq12d 5104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
32	14, 31	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
33	32	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
34		isoun.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦)
35	34	3expb 1120	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → 𝑥𝑅𝑦)
36		isoun.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤)
37	36	3expia 1121	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑤 ∈ 𝐷 → 𝑧𝑆𝑤))
38	37	ralrimiv 3123	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∀𝑤 ∈ 𝐷 𝑧𝑆𝑤)
39	38	ralrimiva 3124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 𝑧𝑆𝑤)
40	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 𝑧𝑆𝑤)
41		f1of 6763	. . . . . . . . . . . . . . . . 17 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴⟶𝐵)
42	3, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)
43	42	ffvelcdmda 7017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) ∈ 𝐵)
44	43	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐻‘𝑥) ∈ 𝐵)
45		f1of 6763	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶⟶𝐷)
46	6, 45	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
47	46	ffvelcdmda 7017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝐺‘𝑦) ∈ 𝐷)
48	47	adantrl 716	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐺‘𝑦) ∈ 𝐷)
49		breq1 5094	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐻‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐻‘𝑥)𝑆𝑤))
50		breq2 5095	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐺‘𝑦) → ((𝐻‘𝑥)𝑆𝑤 ↔ (𝐻‘𝑥)𝑆(𝐺‘𝑦)))
51	49, 50	rspc2v 3588	. . . . . . . . . . . . . 14 ⊢ (((𝐻‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑦) ∈ 𝐷) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 𝑧𝑆𝑤 → (𝐻‘𝑥)𝑆(𝐺‘𝑦)))
52	44, 48, 51	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐷 𝑧𝑆𝑤 → (𝐻‘𝑥)𝑆(𝐺‘𝑦)))
53	40, 52	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝐻‘𝑥)𝑆(𝐺‘𝑦))
54	23	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐻‘𝑥))
55	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐻 Fn 𝐴)
56	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐺 Fn 𝐶)
57	7	anim1i 615	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑦 ∈ 𝐶))
58		fvun2 6914	. . . . . . . . . . . . . 14 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐺 Fn 𝐶 ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐺‘𝑦))
59	55, 56, 57, 58	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐺‘𝑦))
60	59	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐺‘𝑦))
61	53, 54, 60	3brtr4d 5123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))
62	35, 61	2thd 265	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
63	62	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
64	33, 63	jaodan 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
65	12, 64	sylan2b 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
66	65	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝐴 ∪ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))))
67		isoun.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦)
68	67	3expb 1120	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ¬ 𝑥𝑅𝑦)
69		isoun.6	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤)
70	69	3expia 1121	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝑤 ∈ 𝐵 → ¬ 𝑧𝑆𝑤))
71	70	ralrimiv 3123	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ∀𝑤 ∈ 𝐵 ¬ 𝑧𝑆𝑤)
72	71	ralrimiva 3124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∀𝑤 ∈ 𝐵 ¬ 𝑧𝑆𝑤)
73	72	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐷 ∀𝑤 ∈ 𝐵 ¬ 𝑧𝑆𝑤)
74	46	ffvelcdmda 7017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐺‘𝑥) ∈ 𝐷)
75	74	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → (𝐺‘𝑥) ∈ 𝐷)
76	42	ffvelcdmda 7017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐻‘𝑦) ∈ 𝐵)
77	76	adantrl 716	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → (𝐻‘𝑦) ∈ 𝐵)
78		breq1 5094	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐺‘𝑥) → (𝑧𝑆𝑤 ↔ (𝐺‘𝑥)𝑆𝑤))
79	78	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐺‘𝑥) → (¬ 𝑧𝑆𝑤 ↔ ¬ (𝐺‘𝑥)𝑆𝑤))
80		breq2 5095	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝐻‘𝑦) → ((𝐺‘𝑥)𝑆𝑤 ↔ (𝐺‘𝑥)𝑆(𝐻‘𝑦)))
81	80	notbid 318	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐻‘𝑦) → (¬ (𝐺‘𝑥)𝑆𝑤 ↔ ¬ (𝐺‘𝑥)𝑆(𝐻‘𝑦)))
82	79, 81	rspc2v 3588	. . . . . . . . . . . . . 14 ⊢ (((𝐺‘𝑥) ∈ 𝐷 ∧ (𝐻‘𝑦) ∈ 𝐵) → (∀𝑧 ∈ 𝐷 ∀𝑤 ∈ 𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺‘𝑥)𝑆(𝐻‘𝑦)))
83	75, 77, 82	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → (∀𝑧 ∈ 𝐷 ∀𝑤 ∈ 𝐵 ¬ 𝑧𝑆𝑤 → ¬ (𝐺‘𝑥)𝑆(𝐻‘𝑦)))
84	73, 83	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝐺‘𝑥)𝑆(𝐻‘𝑦))
85	16	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐻 Fn 𝐴)
86	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐺 Fn 𝐶)
87	7	anim1i 615	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑥 ∈ 𝐶))
88		fvun2 6914	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 Fn 𝐴 ∧ 𝐺 Fn 𝐶 ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ 𝑥 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐺‘𝑥))
89	85, 86, 87, 88	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐺‘𝑥))
90	89	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐺‘𝑥))
91	29	adantrl 716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐻‘𝑦))
92	90, 91	breq12d 5104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → (((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐻‘𝑦)))
93	84, 92	mtbird 325	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → ¬ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))
94	68, 93	2falsed 376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
95	94	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
96		isorel 7260	. . . . . . . . . . . 12 ⊢ ((𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
97	4, 96	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
98	89	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑥) = (𝐺‘𝑥))
99	59	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐻 ∪ 𝐺)‘𝑦) = (𝐺‘𝑦))
100	98, 99	breq12d 5104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦) ↔ (𝐺‘𝑥)𝑆(𝐺‘𝑦)))
101	97, 100	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
102	101	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
103	95, 102	jaodan 959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
104	12, 103	sylan2b 594	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ (𝐴 ∪ 𝐶)) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
105	104	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑦 ∈ (𝐴 ∪ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))))
106	66, 105	jaodan 959	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶)) → (𝑦 ∈ (𝐴 ∪ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))))
107	11, 106	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ 𝐶)) → (𝑦 ∈ (𝐴 ∪ 𝐶) → (𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))))
108	107	ralrimiv 3123	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ 𝐶)) → ∀𝑦 ∈ (𝐴 ∪ 𝐶)(𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
109	108	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∪ 𝐶)∀𝑦 ∈ (𝐴 ∪ 𝐶)(𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦)))
110		df-isom 6490	. 2 ⊢ ((𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷)) ↔ ((𝐻 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷) ∧ ∀𝑥 ∈ (𝐴 ∪ 𝐶)∀𝑦 ∈ (𝐴 ∪ 𝐶)(𝑥𝑅𝑦 ↔ ((𝐻 ∪ 𝐺)‘𝑥)𝑆((𝐻 ∪ 𝐺)‘𝑦))))
111	10, 109, 110	sylanbrc 583	1 ⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∪ cun 3900   ∩ cin 3901  ∅c0 4283   class class class wbr 5091   Fn wfn 6476  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481   Isom wiso 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490

This theorem is referenced by: (None)