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Theorem f1oiso2 5500
 Description: Any one-to-one onto function determines an isomorphism with an induced relation S. (Contributed by Mario Carneiro, 9-Mar-2013.)
Hypothesis
Ref Expression
f1oiso2.1 S = {x, y ((x B y B) (Hx)R(Hy))}
Assertion
Ref Expression
f1oiso2 (H:A1-1-ontoBH Isom R, S (A, B))
Distinct variable groups:   x,A,y   x,B,y   x,H,y   x,R,y
Allowed substitution hints:   S(x,y)

Proof of Theorem f1oiso2
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oiso2.1 . . 3 S = {x, y ((x B y B) (Hx)R(Hy))}
2 f1ocnvdm 5481 . . . . . . . . 9 ((H:A1-1-ontoB x B) → (Hx) A)
32adantrr 697 . . . . . . . 8 ((H:A1-1-ontoB (x B y B)) → (Hx) A)
433adant3 975 . . . . . . 7 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → (Hx) A)
5 f1ocnvdm 5481 . . . . . . . . . 10 ((H:A1-1-ontoB y B) → (Hy) A)
65adantrl 696 . . . . . . . . 9 ((H:A1-1-ontoB (x B y B)) → (Hy) A)
763adant3 975 . . . . . . . 8 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → (Hy) A)
8 f1ocnvfv2 5477 . . . . . . . . . . 11 ((H:A1-1-ontoB x B) → (H ‘(Hx)) = x)
98eqcomd 2358 . . . . . . . . . 10 ((H:A1-1-ontoB x B) → x = (H ‘(Hx)))
10 f1ocnvfv2 5477 . . . . . . . . . . 11 ((H:A1-1-ontoB y B) → (H ‘(Hy)) = y)
1110eqcomd 2358 . . . . . . . . . 10 ((H:A1-1-ontoB y B) → y = (H ‘(Hy)))
129, 11anim12dan 810 . . . . . . . . 9 ((H:A1-1-ontoB (x B y B)) → (x = (H ‘(Hx)) y = (H ‘(Hy))))
13123adant3 975 . . . . . . . 8 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → (x = (H ‘(Hx)) y = (H ‘(Hy))))
14 simp3 957 . . . . . . . 8 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → (Hx)R(Hy))
15 fveq2 5328 . . . . . . . . . . . 12 (w = (Hy) → (Hw) = (H ‘(Hy)))
1615eqeq2d 2364 . . . . . . . . . . 11 (w = (Hy) → (y = (Hw) ↔ y = (H ‘(Hy))))
1716anbi2d 684 . . . . . . . . . 10 (w = (Hy) → ((x = (H ‘(Hx)) y = (Hw)) ↔ (x = (H ‘(Hx)) y = (H ‘(Hy)))))
18 breq2 4643 . . . . . . . . . 10 (w = (Hy) → ((Hx)Rw ↔ (Hx)R(Hy)))
1917, 18anbi12d 691 . . . . . . . . 9 (w = (Hy) → (((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw) ↔ ((x = (H ‘(Hx)) y = (H ‘(Hy))) (Hx)R(Hy))))
2019rspcev 2955 . . . . . . . 8 (((Hy) A ((x = (H ‘(Hx)) y = (H ‘(Hy))) (Hx)R(Hy))) → w A ((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw))
217, 13, 14, 20syl12anc 1180 . . . . . . 7 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → w A ((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw))
22 fveq2 5328 . . . . . . . . . . . 12 (z = (Hx) → (Hz) = (H ‘(Hx)))
2322eqeq2d 2364 . . . . . . . . . . 11 (z = (Hx) → (x = (Hz) ↔ x = (H ‘(Hx))))
2423anbi1d 685 . . . . . . . . . 10 (z = (Hx) → ((x = (Hz) y = (Hw)) ↔ (x = (H ‘(Hx)) y = (Hw))))
25 breq1 4642 . . . . . . . . . 10 (z = (Hx) → (zRw ↔ (Hx)Rw))
2624, 25anbi12d 691 . . . . . . . . 9 (z = (Hx) → (((x = (Hz) y = (Hw)) zRw) ↔ ((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw)))
2726rexbidv 2635 . . . . . . . 8 (z = (Hx) → (w A ((x = (Hz) y = (Hw)) zRw) ↔ w A ((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw)))
2827rspcev 2955 . . . . . . 7 (((Hx) A w A ((x = (H ‘(Hx)) y = (Hw)) (Hx)Rw)) → z A w A ((x = (Hz) y = (Hw)) zRw))
294, 21, 28syl2anc 642 . . . . . 6 ((H:A1-1-ontoB (x B y B) (Hx)R(Hy)) → z A w A ((x = (Hz) y = (Hw)) zRw))
30293expib 1154 . . . . 5 (H:A1-1-ontoB → (((x B y B) (Hx)R(Hy)) → z A w A ((x = (Hz) y = (Hw)) zRw)))
31 simp3ll 1026 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → x = (Hz))
32 simp1 955 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → H:A1-1-ontoB)
33 simp2l 981 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → z A)
34 f1of 5287 . . . . . . . . . . 11 (H:A1-1-ontoBH:A–→B)
35 ffvelrn 5415 . . . . . . . . . . 11 ((H:A–→B z A) → (Hz) B)
3634, 35sylan 457 . . . . . . . . . 10 ((H:A1-1-ontoB z A) → (Hz) B)
3732, 33, 36syl2anc 642 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hz) B)
3831, 37eqeltrd 2427 . . . . . . . 8 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → x B)
39 simp3lr 1027 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → y = (Hw))
40 simp2r 982 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → w A)
41 ffvelrn 5415 . . . . . . . . . . 11 ((H:A–→B w A) → (Hw) B)
4234, 41sylan 457 . . . . . . . . . 10 ((H:A1-1-ontoB w A) → (Hw) B)
4332, 40, 42syl2anc 642 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hw) B)
4439, 43eqeltrd 2427 . . . . . . . 8 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → y B)
45 simp3r 984 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → zRw)
4631eqcomd 2358 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hz) = x)
47 f1ocnvfv 5478 . . . . . . . . . . 11 ((H:A1-1-ontoB z A) → ((Hz) = x → (Hx) = z))
4832, 33, 47syl2anc 642 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → ((Hz) = x → (Hx) = z))
4946, 48mpd 14 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hx) = z)
5039eqcomd 2358 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hw) = y)
51 f1ocnvfv 5478 . . . . . . . . . . 11 ((H:A1-1-ontoB w A) → ((Hw) = y → (Hy) = w))
5232, 40, 51syl2anc 642 . . . . . . . . . 10 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → ((Hw) = y → (Hy) = w))
5350, 52mpd 14 . . . . . . . . 9 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hy) = w)
5445, 49, 533brtr4d 4669 . . . . . . . 8 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → (Hx)R(Hy))
5538, 44, 54jca31 520 . . . . . . 7 ((H:A1-1-ontoB (z A w A) ((x = (Hz) y = (Hw)) zRw)) → ((x B y B) (Hx)R(Hy)))
56553exp 1150 . . . . . 6 (H:A1-1-ontoB → ((z A w A) → (((x = (Hz) y = (Hw)) zRw) → ((x B y B) (Hx)R(Hy)))))
5756rexlimdvv 2744 . . . . 5 (H:A1-1-ontoB → (z A w A ((x = (Hz) y = (Hw)) zRw) → ((x B y B) (Hx)R(Hy))))
5830, 57impbid 183 . . . 4 (H:A1-1-ontoB → (((x B y B) (Hx)R(Hy)) ↔ z A w A ((x = (Hz) y = (Hw)) zRw)))
5958opabbidv 4625 . . 3 (H:A1-1-ontoB → {x, y ((x B y B) (Hx)R(Hy))} = {x, y z A w A ((x = (Hz) y = (Hw)) zRw)})
601, 59syl5eq 2397 . 2 (H:A1-1-ontoBS = {x, y z A w A ((x = (Hz) y = (Hw)) zRw)})
61 f1oiso 5499 . 2 ((H:A1-1-ontoB S = {x, y z A w A ((x = (Hz) y = (Hw)) zRw)}) → H Isom R, S (A, B))
6260, 61mpdan 649 1 (H:A1-1-ontoBH Isom R, S (A, B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {copab 4622   class class class wbr 4639  ◡ccnv 4771  –→wf 4777  –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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