Theorem ordiso2 8978

Description: Generalize ordiso 8979 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
ordiso2	⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Proof of Theorem ordiso2

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordsson 7503	. . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
2	1	3ad2ant2 1130	. . . . 5 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 ⊆ On)
3	2	sseld 3965	. . . 4 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
4		eleq1w 2895	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5		fveq2 6669	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
6		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
7	5, 6	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑦) = 𝑦))
8	4, 7	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥) ↔ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)))
9	8	imbi2d 343	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)) ↔ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦))))
10		r19.21v 3175	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) ↔ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)))
11		ordelss 6206	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
12	11	3ad2antl2 1182	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
13	12	sselda 3966	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
14		pm5.5 364	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ (𝐹‘𝑦) = 𝑦))
15	13, 14	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ (𝐹‘𝑦) = 𝑦))
16	15	ralbidva 3196	. . . . . . . . . . . 12 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦))
17		isof1o 7075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
18	17	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
19	18	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹:𝐴–1-1-onto→𝐵)
20		simpll3 1210	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → Ord 𝐵)
21		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ (𝐹‘𝑥))
22		f1of 6614	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
23	17, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴⟶𝐵)
24	23	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹:𝐴⟶𝐵)
25	24	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹:𝐴⟶𝐵)
26		simplrl 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑥 ∈ 𝐴)
27	25, 26	ffvelrnd 6851	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵)
28	21, 27	jca 514	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝑧 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵))
29		ordtr1 6233	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝐵 → ((𝑧 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑧 ∈ 𝐵))
30	20, 28, 29	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ 𝐵)
31		f1ocnvfv2 7033	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
32	19, 30, 31	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧)
33	32, 21	eqeltrd 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥))
34		simpll1 1208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹 Isom E , E (𝐴, 𝐵))
35		f1ocnv 6626	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
36		f1of 6614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
37	19, 35, 36	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ◡𝐹:𝐵⟶𝐴)
38	37, 30	ffvelrnd 6851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (◡𝐹‘𝑧) ∈ 𝐴)
39		isorel 7078	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ ((◡𝐹‘𝑧) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((◡𝐹‘𝑧) E 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥)))
40	34, 38, 26, 39	syl12anc 834	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ((◡𝐹‘𝑧) E 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥)))
41		epel 5468	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((◡𝐹‘𝑧) E 𝑥 ↔ (◡𝐹‘𝑧) ∈ 𝑥)
42		fvex 6682	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹‘𝑥) ∈ V
43	42	epeli 5467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥) ↔ (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥))
44	40, 41, 43	3bitr3g 315	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ((◡𝐹‘𝑧) ∈ 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥)))
45	33, 44	mpbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (◡𝐹‘𝑧) ∈ 𝑥)
46		simplrr 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)
47		fveq2 6669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑧)))
48		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (◡𝐹‘𝑧) → 𝑦 = (◡𝐹‘𝑧))
49	47, 48	eqeq12d 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝐹‘𝑦) = 𝑦 ↔ (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧)))
50	49	rspcv 3617	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑧) ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦 → (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧)))
51	45, 46, 50	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧))
52	32, 51	eqtr3d 2858	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 = (◡𝐹‘𝑧))
53	52, 45	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ 𝑥)
54		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)
55		fveq2 6669	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
56		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
57	55, 56	eqeq12d 2837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = 𝑦 ↔ (𝐹‘𝑧) = 𝑧))
58	57	rspccva 3621	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦 ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) = 𝑧)
59	54, 58	sylan 582	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) = 𝑧)
60		epel 5468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥)
61	60	biimpri 230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → 𝑧 E 𝑥)
62	61	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 E 𝑥)
63		simpll1 1208	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝐹 Isom E , E (𝐴, 𝐵))
64		simpl2 1188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → Ord 𝐴)
65		simprl 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → 𝑥 ∈ 𝐴)
66	64, 65, 11	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → 𝑥 ⊆ 𝐴)
67	66	sselda 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝐴)
68		simplrl 775	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝐴)
69		isorel 7078	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑧 E 𝑥 ↔ (𝐹‘𝑧) E (𝐹‘𝑥)))
70	63, 67, 68, 69	syl12anc 834	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝑧 E 𝑥 ↔ (𝐹‘𝑧) E (𝐹‘𝑥)))
71	62, 70	mpbid 234	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) E (𝐹‘𝑥))
72	42	epeli 5467	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑧) E (𝐹‘𝑥) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑥))
73	71, 72	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) ∈ (𝐹‘𝑥))
74	59, 73	eqeltrrd 2914	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝐹‘𝑥))
75	53, 74	impbida 799	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ 𝑥))
76	75	eqrdv 2819	. . . . . . . . . . . . 13 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → (𝐹‘𝑥) = 𝑥)
77	76	expr 459	. . . . . . . . . . . 12 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦 → (𝐹‘𝑥) = 𝑥))
78	16, 77	sylbid 242	. . . . . . . . . . 11 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝐹‘𝑥) = 𝑥))
79	78	ex 415	. . . . . . . . . 10 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝐹‘𝑥) = 𝑥)))
80	79	com23 86	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)))
81	80	a2i 14	. . . . . . . 8 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)))
82	81	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ On → (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))))
83	10, 82	syl5bi 244	. . . . . 6 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))))
84	9, 83	tfis2 7570	. . . . 5 ⊢ (𝑥 ∈ On → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)))
85	84	com3l 89	. . . 4 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ On → (𝐹‘𝑥) = 𝑥)))
86	3, 85	mpdd 43	. . 3 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))
87	86	ralrimiv 3181	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥)
88		fveq2 6669	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
89		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
90	88, 89	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑧) = 𝑧))
91	90	rspccva 3621	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝑧)
92	91	adantll 712	. . . . . 6 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝑧)
93	23	ffvelrnda 6850	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
94	93	3ad2antl1 1181	. . . . . . 7 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
95	94	adantlr 713	. . . . . 6 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
96	92, 95	eqeltrrd 2914	. . . . 5 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐵)
97	96	ex 415	. . . 4 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
98		simpl1 1187	. . . . . . . 8 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐹 Isom E , E (𝐴, 𝐵))
99		f1ofo 6621	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
100		forn 6592	. . . . . . . . 9 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
101	17, 99, 100	3syl 18	. . . . . . . 8 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ran 𝐹 = 𝐵)
102	98, 101	syl 17	. . . . . . 7 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → ran 𝐹 = 𝐵)
103	102	eleq2d 2898	. . . . . 6 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐵))
104		f1ofn 6615	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
105	17, 104	syl 17	. . . . . . . . 9 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹 Fn 𝐴)
106	105	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹 Fn 𝐴)
107	106	adantr 483	. . . . . . 7 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐹 Fn 𝐴)
108		fvelrnb 6725	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧))
109	107, 108	syl 17	. . . . . 6 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧))
110	103, 109	bitr3d 283	. . . . 5 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐵 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧))
111		fveq2 6669	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
112		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
113	111, 112	eqeq12d 2837	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑤) = 𝑤))
114	113	rspcv 3617	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝐹‘𝑤) = 𝑤))
115	114	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝐹‘𝑤) = 𝑤)))
116		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑧)
117		simpl 485	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑤)
118	116, 117	eqtr3d 2858	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → 𝑧 = 𝑤)
119	118	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑧 = 𝑤)
120		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑤 ∈ 𝐴)
121	119, 120	eqeltrd 2913	. . . . . . . . . 10 ⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑧 ∈ 𝐴)
122	121	exp43 439	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑤 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴))))
123	115, 122	syldd 72	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴))))
124	123	com23 86	. . . . . . 7 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴))))
125	124	imp 409	. . . . . 6 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴)))
126	125	rexlimdv 3283	. . . . 5 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴))
127	110, 126	sylbid 242	. . . 4 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴))
128	97, 127	impbid 214	. . 3 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
129	128	eqrdv 2819	. 2 ⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐴 = 𝐵)
130	87, 129	mpdan 685	1 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935   class class class wbr 5065   E cep 5463  ^◡ccnv 5553  ran crn 5555  Ord word 6189  Oncon0 6190   Fn wfn 6349  ⟶wf 6350  –onto→wfo 6352  –1-1-onto→wf1o 6353  ‘cfv 6354   Isom wiso 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363

This theorem is referenced by:  ordiso  8979  oieu  9002  oiid  9004