Theorem isomin 5496

Description: Isomorphisms preserve minimal elements. Note that (^◡R “ {D}) is Takeuti and Zaring's idiom for the initial segment {x ∣ xRD}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by set.mm contributors, 19-Apr-2004.)

Assertion

Ref	Expression
isomin	⊢ ((H Isom R, S (A, B) ∧ (C ⊆ A ∧ D ∈ A)) → ((C ∩ (◡R “ {D})) = ∅ ↔ ((H “ C) ∩ (◡S “ {(H ‘D)})) = ∅))

Proof of Theorem isomin

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

Step	Hyp	Ref	Expression
1		ssel2 3268	. . . . . . . 8 ⊢ ((C ⊆ A ∧ y ∈ C) → y ∈ A)
2	1	anim1i 551	. . . . . . 7 ⊢ (((C ⊆ A ∧ y ∈ C) ∧ D ∈ A) → (y ∈ A ∧ D ∈ A))
3	2	an32s 779	. . . . . 6 ⊢ (((C ⊆ A ∧ D ∈ A) ∧ y ∈ C) → (y ∈ A ∧ D ∈ A))
4		isorel 5489	. . . . . . 7 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → (yRD ↔ (H ‘y)S(H ‘D)))
5		fvex 5339	. . . . . . . . 9 ⊢ (H ‘y) ∈ V
6		breq1 4642	. . . . . . . . 9 ⊢ (x = (H ‘y) → (xS(H ‘D) ↔ (H ‘y)S(H ‘D)))
7	5, 6	ceqsexv 2894	. . . . . . . 8 ⊢ (∃x(x = (H ‘y) ∧ xS(H ‘D)) ↔ (H ‘y)S(H ‘D))
8		eqcom 2355	. . . . . . . . . . 11 ⊢ (x = (H ‘y) ↔ (H ‘y) = x)
9		isof1o 5488	. . . . . . . . . . . . 13 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)
10		f1ofn 5288	. . . . . . . . . . . . 13 ⊢ (H:A–1-1-onto→B → H Fn A)
11	9, 10	syl 15	. . . . . . . . . . . 12 ⊢ (H Isom R, S (A, B) → H Fn A)
12		simpl 443	. . . . . . . . . . . 12 ⊢ ((y ∈ A ∧ D ∈ A) → y ∈ A)
13		fnbrfvb 5358	. . . . . . . . . . . 12 ⊢ ((H Fn A ∧ y ∈ A) → ((H ‘y) = x ↔ yHx))
14	11, 12, 13	syl2an 463	. . . . . . . . . . 11 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → ((H ‘y) = x ↔ yHx))
15	8, 14	syl5bb 248	. . . . . . . . . 10 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → (x = (H ‘y) ↔ yHx))
16	15	anbi1d 685	. . . . . . . . 9 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → ((x = (H ‘y) ∧ xS(H ‘D)) ↔ (yHx ∧ xS(H ‘D))))
17	16	exbidv 1626	. . . . . . . 8 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → (∃x(x = (H ‘y) ∧ xS(H ‘D)) ↔ ∃x(yHx ∧ xS(H ‘D))))
18	7, 17	syl5bbr 250	. . . . . . 7 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → ((H ‘y)S(H ‘D) ↔ ∃x(yHx ∧ xS(H ‘D))))
19	4, 18	bitrd 244	. . . . . 6 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ D ∈ A)) → (yRD ↔ ∃x(yHx ∧ xS(H ‘D))))
20	3, 19	sylan2 460	. . . . 5 ⊢ ((H Isom R, S (A, B) ∧ ((C ⊆ A ∧ D ∈ A) ∧ y ∈ C)) → (yRD ↔ ∃x(yHx ∧ xS(H ‘D))))
21	20	anassrs 629	. . . 4 ⊢ (((H Isom R, S (A, B) ∧ (C ⊆ A ∧ D ∈ A)) ∧ y ∈ C) → (yRD ↔ ∃x(yHx ∧ xS(H ‘D))))
22	21	rexbidva 2631	. . 3 ⊢ ((H Isom R, S (A, B) ∧ (C ⊆ A ∧ D ∈ A)) → (∃y ∈ C yRD ↔ ∃y ∈ C ∃x(yHx ∧ xS(H ‘D))))
23		elin 3219	. . . . . 6 ⊢ (y ∈ (C ∩ (◡R “ {D})) ↔ (y ∈ C ∧ y ∈ (◡R “ {D})))
24		eliniseg 5020	. . . . . . 7 ⊢ (y ∈ (◡R “ {D}) ↔ yRD)
25	24	anbi2i 675	. . . . . 6 ⊢ ((y ∈ C ∧ y ∈ (◡R “ {D})) ↔ (y ∈ C ∧ yRD))
26	23, 25	bitri 240	. . . . 5 ⊢ (y ∈ (C ∩ (◡R “ {D})) ↔ (y ∈ C ∧ yRD))
27	26	exbii 1582	. . . 4 ⊢ (∃y y ∈ (C ∩ (◡R “ {D})) ↔ ∃y(y ∈ C ∧ yRD))
28		neq0 3560	. . . 4 ⊢ (¬ (C ∩ (◡R “ {D})) = ∅ ↔ ∃y y ∈ (C ∩ (◡R “ {D})))
29		df-rex 2620	. . . 4 ⊢ (∃y ∈ C yRD ↔ ∃y(y ∈ C ∧ yRD))
30	27, 28, 29	3bitr4i 268	. . 3 ⊢ (¬ (C ∩ (◡R “ {D})) = ∅ ↔ ∃y ∈ C yRD)
31		elima 4754	. . . . . . 7 ⊢ (x ∈ (H “ C) ↔ ∃y ∈ C yHx)
32		eliniseg 5020	. . . . . . 7 ⊢ (x ∈ (◡S “ {(H ‘D)}) ↔ xS(H ‘D))
33	31, 32	anbi12i 678	. . . . . 6 ⊢ ((x ∈ (H “ C) ∧ x ∈ (◡S “ {(H ‘D)})) ↔ (∃y ∈ C yHx ∧ xS(H ‘D)))
34		elin 3219	. . . . . 6 ⊢ (x ∈ ((H “ C) ∩ (◡S “ {(H ‘D)})) ↔ (x ∈ (H “ C) ∧ x ∈ (◡S “ {(H ‘D)})))
35		r19.41v 2764	. . . . . 6 ⊢ (∃y ∈ C (yHx ∧ xS(H ‘D)) ↔ (∃y ∈ C yHx ∧ xS(H ‘D)))
36	33, 34, 35	3bitr4i 268	. . . . 5 ⊢ (x ∈ ((H “ C) ∩ (◡S “ {(H ‘D)})) ↔ ∃y ∈ C (yHx ∧ xS(H ‘D)))
37	36	exbii 1582	. . . 4 ⊢ (∃x x ∈ ((H “ C) ∩ (◡S “ {(H ‘D)})) ↔ ∃x∃y ∈ C (yHx ∧ xS(H ‘D)))
38		neq0 3560	. . . 4 ⊢ (¬ ((H “ C) ∩ (◡S “ {(H ‘D)})) = ∅ ↔ ∃x x ∈ ((H “ C) ∩ (◡S “ {(H ‘D)})))
39		rexcom4 2878	. . . 4 ⊢ (∃y ∈ C ∃x(yHx ∧ xS(H ‘D)) ↔ ∃x∃y ∈ C (yHx ∧ xS(H ‘D)))
40	37, 38, 39	3bitr4i 268	. . 3 ⊢ (¬ ((H “ C) ∩ (◡S “ {(H ‘D)})) = ∅ ↔ ∃y ∈ C ∃x(yHx ∧ xS(H ‘D)))
41	22, 30, 40	3bitr4g 279	. 2 ⊢ ((H Isom R, S (A, B) ∧ (C ⊆ A ∧ D ∈ A)) → (¬ (C ∩ (◡R “ {D})) = ∅ ↔ ¬ ((H “ C) ∩ (◡S “ {(H ‘D)})) = ∅))
42	41	con4bid 284	1 ⊢ ((H Isom R, S (A, B) ∧ (C ⊆ A ∧ D ∈ A)) → ((C ∩ (◡R “ {D})) = ∅ ↔ ((H “ C) ∩ (◡S “ {(H ‘D)})) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737   class class class wbr 4639   “ cima 4722  ^◡ccnv 4771   Fn wfn 4776  –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-f1o 4794  df-fv 4795  df-iso 4796

This theorem is referenced by: (None)