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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 “ {(HX)}))
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.
StepHypRef Expression
1 isof1o 5488 . . . . . 6 (H Isom R, S (A, B) → H:A1-1-ontoB)
2 f1of1 5286 . . . . . 6 (H:A1-1-ontoBH:A1-1B)
31, 2syl 15 . . . . 5 (H Isom R, S (A, B) → H:A1-1B)
43adantr 451 . . . 4 ((H Isom R, S (A, B) X A) → H:A1-1B)
5 isoini2.1 . . . . 5 C = (A ∩ (R “ {X}))
6 inss1 3475 . . . . 5 (A ∩ (R “ {X})) A
75, 6eqsstri 3301 . . . 4 C A
8 f1ores 5300 . . . 4 ((H:A1-1B C A) → (H C):C1-1-onto→(HC))
94, 7, 8sylancl 643 . . 3 ((H Isom R, S (A, B) X A) → (H C):C1-1-onto→(HC))
10 isoini 5497 . . . . 5 ((H Isom R, S (A, B) X A) → (H “ (A ∩ (R “ {X}))) = (B ∩ (S “ {(HX)})))
115imaeq2i 4940 . . . . 5 (HC) = (H “ (A ∩ (R “ {X})))
12 isoini2.2 . . . . 5 D = (B ∩ (S “ {(HX)}))
1310, 11, 123eqtr4g 2410 . . . 4 ((H Isom R, S (A, B) X A) → (HC) = D)
14 f1oeq3 5283 . . . 4 ((HC) = D → ((H C):C1-1-onto→(HC) ↔ (H C):C1-1-ontoD))
1513, 14syl 15 . . 3 ((H Isom R, S (A, B) X A) → ((H C):C1-1-onto→(HC) ↔ (H C):C1-1-ontoD))
169, 15mpbid 201 . 2 ((H Isom R, S (A, B) X A) → (H C):C1-1-ontoD)
17 df-iso 4796 . . . . . . 7 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
1817simprbi 450 . . . . . 6 (H Isom R, S (A, B) → x A y A (xRy ↔ (Hx)S(Hy)))
1918adantr 451 . . . . 5 ((H Isom R, S (A, B) X A) → x A y A (xRy ↔ (Hx)S(Hy)))
20 ssralv 3330 . . . . . 6 (C A → (y A (xRy ↔ (Hx)S(Hy)) → y C (xRy ↔ (Hx)S(Hy))))
2120ralimdv 2693 . . . . 5 (C A → (x A y A (xRy ↔ (Hx)S(Hy)) → x A y C (xRy ↔ (Hx)S(Hy))))
227, 19, 21mpsyl 59 . . . 4 ((H Isom R, S (A, B) X A) → x A y C (xRy ↔ (Hx)S(Hy)))
23 ssralv 3330 . . . 4 (C A → (x A y C (xRy ↔ (Hx)S(Hy)) → x C y C (xRy ↔ (Hx)S(Hy))))
247, 22, 23mpsyl 59 . . 3 ((H Isom R, S (A, B) X A) → x C y C (xRy ↔ (Hx)S(Hy)))
25 fvres 5342 . . . . . . 7 (x C → ((H C) ‘x) = (Hx))
26 fvres 5342 . . . . . . 7 (y C → ((H C) ‘y) = (Hy))
2725, 26breqan12d 4654 . . . . . 6 ((x C y C) → (((H C) ‘x)S((H C) ‘y) ↔ (Hx)S(Hy)))
2827bibi2d 309 . . . . 5 ((x C y C) → ((xRy ↔ ((H C) ‘x)S((H C) ‘y)) ↔ (xRy ↔ (Hx)S(Hy))))
2928ralbidva 2630 . . . 4 (x C → (y C (xRy ↔ ((H C) ‘x)S((H C) ‘y)) ↔ y C (xRy ↔ (Hx)S(Hy))))
3029ralbiia 2646 . . 3 (x C y C (xRy ↔ ((H C) ‘x)S((H C) ‘y)) ↔ x C y C (xRy ↔ (Hx)S(Hy)))
3124, 30sylibr 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):C1-1-ontoD x C y C (xRy ↔ ((H C) ‘x)S((H C) ‘y))))
3316, 31, 32sylanbrc 645 1 ((H Isom R, S (A, B) X A) → (H C) Isom R, S (C, D))
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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)
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