Theorem isoini2 5498

Description: Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Hypotheses

Ref	Expression
isoini2.1	⊢ C = (A ∩ (◡R “ {X}))
isoini2.2	⊢ D = (B ∩ (◡S “ {(H ‘X)}))

Assertion

Ref	Expression
isoini2	⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H ↾ C) Isom R, S (C, D))

Proof of Theorem isoini2

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

Step	Hyp	Ref	Expression
1		isof1o 5488	. . . . . 6 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)
2		f1of1 5286	. . . . . 6 ⊢ (H:A–1-1-onto→B → H:A–1-1→B)
3	1, 2	syl 15	. . . . 5 ⊢ (H Isom R, S (A, B) → H:A–1-1→B)
4	3	adantr 451	. . . 4 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → H:A–1-1→B)
5		isoini2.1	. . . . 5 ⊢ C = (A ∩ (◡R “ {X}))
6		inss1 3475	. . . . 5 ⊢ (A ∩ (◡R “ {X})) ⊆ A
7	5, 6	eqsstri 3301	. . . 4 ⊢ C ⊆ A
8		f1ores 5300	. . . 4 ⊢ ((H:A–1-1→B ∧ C ⊆ A) → (H ↾ C):C–1-1-onto→(H “ C))
9	4, 7, 8	sylancl 643	. . 3 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H ↾ C):C–1-1-onto→(H “ C))
10		isoini 5497	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H “ (A ∩ (◡R “ {X}))) = (B ∩ (◡S “ {(H ‘X)})))
11	5	imaeq2i 4940	. . . . 5 ⊢ (H “ C) = (H “ (A ∩ (◡R “ {X})))
12		isoini2.2	. . . . 5 ⊢ D = (B ∩ (◡S “ {(H ‘X)}))
13	10, 11, 12	3eqtr4g 2410	. . . 4 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H “ C) = D)
14		f1oeq3 5283	. . . 4 ⊢ ((H “ C) = D → ((H ↾ C):C–1-1-onto→(H “ C) ↔ (H ↾ C):C–1-1-onto→D))
15	13, 14	syl 15	. . 3 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → ((H ↾ C):C–1-1-onto→(H “ C) ↔ (H ↾ C):C–1-1-onto→D))
16	9, 15	mpbid 201	. 2 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H ↾ C):C–1-1-onto→D)
17		df-iso 4796	. . . . . . 7 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
18	17	simprbi 450	. . . . . 6 ⊢ (H Isom R, S (A, B) → ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))
19	18	adantr 451	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))
20		ssralv 3330	. . . . . 6 ⊢ (C ⊆ A → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) → ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
21	20	ralimdv 2693	. . . . 5 ⊢ (C ⊆ A → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) → ∀x ∈ A ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
22	7, 19, 21	mpsyl 59	. . . 4 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → ∀x ∈ A ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y)))
23		ssralv 3330	. . . 4 ⊢ (C ⊆ A → (∀x ∈ A ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y)) → ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
24	7, 22, 23	mpsyl 59	. . 3 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y)))
25		fvres 5342	. . . . . . 7 ⊢ (x ∈ C → ((H ↾ C) ‘x) = (H ‘x))
26		fvres 5342	. . . . . . 7 ⊢ (y ∈ C → ((H ↾ C) ‘y) = (H ‘y))
27	25, 26	breqan12d 4654	. . . . . 6 ⊢ ((x ∈ C ∧ y ∈ C) → (((H ↾ C) ‘x)S((H ↾ C) ‘y) ↔ (H ‘x)S(H ‘y)))
28	27	bibi2d 309	. . . . 5 ⊢ ((x ∈ C ∧ y ∈ C) → ((xRy ↔ ((H ↾ C) ‘x)S((H ↾ C) ‘y)) ↔ (xRy ↔ (H ‘x)S(H ‘y))))
29	28	ralbidva 2630	. . . 4 ⊢ (x ∈ C → (∀y ∈ C (xRy ↔ ((H ↾ C) ‘x)S((H ↾ C) ‘y)) ↔ ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
30	29	ralbiia 2646	. . 3 ⊢ (∀x ∈ C ∀y ∈ C (xRy ↔ ((H ↾ C) ‘x)S((H ↾ C) ‘y)) ↔ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y)))
31	24, 30	sylibr 203	. 2 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → ∀x ∈ C ∀y ∈ C (xRy ↔ ((H ↾ C) ‘x)S((H ↾ C) ‘y)))
32		df-iso 4796	. 2 ⊢ ((H ↾ C) Isom R, S (C, D) ↔ ((H ↾ C):C–1-1-onto→D ∧ ∀x ∈ C ∀y ∈ C (xRy ↔ ((H ↾ C) ‘x)S((H ↾ C) ‘y))))
33	16, 31, 32	sylanbrc 645	1 ⊢ ((H Isom R, S (A, B) ∧ X ∈ A) → (H ↾ C) Isom R, S (C, D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208   ⊆ wss 3257  {csn 3737   class class class wbr 4639   “ cima 4722  ^◡ccnv 4771   ↾ cres 4774  –1-1→wf1 4778  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-iso 4796

This theorem is referenced by: (None)