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Theorem sbthlem1 6203
Description: Lemma for sbth 6206. Set up similarity with a range. Theorem XI.1.14 of [Rosser] p. 350. (Contributed by SF, 11-Mar-2015.)
Hypotheses
Ref Expression
sbthlem1.1 R V
sbthlem1.2 X V
sbthlem1.3 G = Clos1 ((X ran R), R)
sbthlem1.4 A = (XG)
sbthlem1.5 B = (X G)
sbthlem1.6 C = (ran RG)
sbthlem1.7 D = (ran R G)
Assertion
Ref Expression
sbthlem1 (((Fun R Fun R) (X dom R ran R X)) → ran RX)

Proof of Theorem sbthlem1
StepHypRef Expression
1 df-f1 4792 . . . . . . . . 9 (R:dom R1-1→ran R ↔ (R:dom R–→ran R Fun R))
2 ssid 3290 . . . . . . . . . . . 12 ran R ran R
3 df-f 4791 . . . . . . . . . . . 12 (R:dom R–→ran R ↔ (R Fn dom R ran R ran R))
42, 3mpbiran2 885 . . . . . . . . . . 11 (R:dom R–→ran RR Fn dom R)
5 funfn 5136 . . . . . . . . . . 11 (Fun RR Fn dom R)
64, 5bitr4i 243 . . . . . . . . . 10 (R:dom R–→ran R ↔ Fun R)
76anbi1i 676 . . . . . . . . 9 ((R:dom R–→ran R Fun R) ↔ (Fun R Fun R))
81, 7bitri 240 . . . . . . . 8 (R:dom R1-1→ran R ↔ (Fun R Fun R))
98biimpri 197 . . . . . . 7 ((Fun R Fun R) → R:dom R1-1→ran R)
10 sbthlem1.4 . . . . . . . . 9 A = (XG)
11 inss1 3475 . . . . . . . . . 10 (XG) X
12 sstr 3280 . . . . . . . . . 10 (((XG) X X dom R) → (XG) dom R)
1311, 12mpan 651 . . . . . . . . 9 (X dom R → (XG) dom R)
1410, 13syl5eqss 3315 . . . . . . . 8 (X dom RA dom R)
1514adantr 451 . . . . . . 7 ((X dom R ran R X) → A dom R)
16 f1ores 5300 . . . . . . 7 ((R:dom R1-1→ran R A dom R) → (R A):A1-1-onto→(RA))
179, 15, 16syl2an 463 . . . . . 6 (((Fun R Fun R) (X dom R ran R X)) → (R A):A1-1-onto→(RA))
18 sbthlem1.1 . . . . . . . 8 R V
19 sbthlem1.2 . . . . . . . . . 10 X V
20 sbthlem1.3 . . . . . . . . . . 11 G = Clos1 ((X ran R), R)
2118rnex 5107 . . . . . . . . . . . . 13 ran R V
2219, 21difex 4107 . . . . . . . . . . . 12 (X ran R) V
2322, 18clos1ex 5876 . . . . . . . . . . 11 Clos1 ((X ran R), R) V
2420, 23eqeltri 2423 . . . . . . . . . 10 G V
2519, 24inex 4105 . . . . . . . . 9 (XG) V
2610, 25eqeltri 2423 . . . . . . . 8 A V
2718, 26resex 5117 . . . . . . 7 (R A) V
2827f1oen 6033 . . . . . 6 ((R A):A1-1-onto→(RA) → A ≈ (RA))
2917, 28syl 15 . . . . 5 (((Fun R Fun R) (X dom R ran R X)) → A ≈ (RA))
30 sbthlem1.6 . . . . . . . . 9 C = (ran RG)
3122, 18, 20clos1baseima 5883 . . . . . . . . . 10 G = ((X ran R) ∪ (RG))
3231ineq2i 3454 . . . . . . . . 9 (ran RG) = (ran R ∩ ((X ran R) ∪ (RG)))
33 indi 3501 . . . . . . . . . 10 (ran R ∩ ((X ran R) ∪ (RG))) = ((ran R ∩ (X ran R)) ∪ (ran R ∩ (RG)))
34 disjdif 3622 . . . . . . . . . . 11 (ran R ∩ (X ran R)) =
3534uneq1i 3414 . . . . . . . . . 10 ((ran R ∩ (X ran R)) ∪ (ran R ∩ (RG))) = ( ∪ (ran R ∩ (RG)))
36 uncom 3408 . . . . . . . . . . 11 ( ∪ (ran R ∩ (RG))) = ((ran R ∩ (RG)) ∪ )
37 un0 3575 . . . . . . . . . . 11 ((ran R ∩ (RG)) ∪ ) = (ran R ∩ (RG))
3836, 37eqtri 2373 . . . . . . . . . 10 ( ∪ (ran R ∩ (RG))) = (ran R ∩ (RG))
3933, 35, 383eqtri 2377 . . . . . . . . 9 (ran R ∩ ((X ran R) ∪ (RG))) = (ran R ∩ (RG))
4030, 32, 393eqtri 2377 . . . . . . . 8 C = (ran R ∩ (RG))
41 inss2 3476 . . . . . . . . 9 (ran R ∩ (RG)) (RG)
4241a1i 10 . . . . . . . 8 (((Fun R Fun R) (X dom R ran R X)) → (ran R ∩ (RG)) (RG))
4340, 42syl5eqss 3315 . . . . . . 7 (((Fun R Fun R) (X dom R ran R X)) → C (RG))
44 imassrn 5009 . . . . . . . . . . . . . 14 (RG) ran R
45 simprr 733 . . . . . . . . . . . . . 14 (((Fun R Fun R) (X dom R ran R X)) → ran R X)
4644, 45syl5ss 3283 . . . . . . . . . . . . 13 (((Fun R Fun R) (X dom R ran R X)) → (RG) X)
47 difss 3393 . . . . . . . . . . . . 13 (X ran R) X
4846, 47jctil 523 . . . . . . . . . . . 12 (((Fun R Fun R) (X dom R ran R X)) → ((X ran R) X (RG) X))
49 unss 3437 . . . . . . . . . . . 12 (((X ran R) X (RG) X) ↔ ((X ran R) ∪ (RG)) X)
5048, 49sylib 188 . . . . . . . . . . 11 (((Fun R Fun R) (X dom R ran R X)) → ((X ran R) ∪ (RG)) X)
5131, 50syl5eqss 3315 . . . . . . . . . 10 (((Fun R Fun R) (X dom R ran R X)) → G X)
52 sseqin2 3474 . . . . . . . . . 10 (G X ↔ (XG) = G)
5351, 52sylib 188 . . . . . . . . 9 (((Fun R Fun R) (X dom R ran R X)) → (XG) = G)
5410, 53syl5eq 2397 . . . . . . . 8 (((Fun R Fun R) (X dom R ran R X)) → A = G)
5554imaeq2d 4942 . . . . . . 7 (((Fun R Fun R) (X dom R ran R X)) → (RA) = (RG))
5643, 55sseqtr4d 3308 . . . . . 6 (((Fun R Fun R) (X dom R ran R X)) → C (RA))
57 ssun2 3427 . . . . . . . . . . 11 (RG) ((X ran R) ∪ (RG))
5857, 31sseqtr4i 3304 . . . . . . . . . 10 (RG) G
5955sseq1d 3298 . . . . . . . . . 10 (((Fun R Fun R) (X dom R ran R X)) → ((RA) G ↔ (RG) G))
6058, 59mpbiri 224 . . . . . . . . 9 (((Fun R Fun R) (X dom R ran R X)) → (RA) G)
61 imassrn 5009 . . . . . . . . 9 (RA) ran R
6260, 61jctil 523 . . . . . . . 8 (((Fun R Fun R) (X dom R ran R X)) → ((RA) ran R (RA) G))
63 ssin 3477 . . . . . . . 8 (((RA) ran R (RA) G) ↔ (RA) (ran RG))
6462, 63sylib 188 . . . . . . 7 (((Fun R Fun R) (X dom R ran R X)) → (RA) (ran RG))
6564, 30syl6sseqr 3318 . . . . . 6 (((Fun R Fun R) (X dom R ran R X)) → (RA) C)
6656, 65eqssd 3289 . . . . 5 (((Fun R Fun R) (X dom R ran R X)) → C = (RA))
6729, 66breqtrrd 4665 . . . 4 (((Fun R Fun R) (X dom R ran R X)) → AC)
68 sbthlem1.5 . . . . . . 7 B = (X G)
6919, 24difex 4107 . . . . . . 7 (X G) V
7068, 69eqeltri 2423 . . . . . 6 B V
7170enrflx 6035 . . . . 5 BB
72 difsscompl 3549 . . . . . . . . . 10 (X G) G
7372a1i 10 . . . . . . . . 9 (((Fun R Fun R) (X dom R ran R X)) → (X G) G)
74 df-dif 3215 . . . . . . . . . . 11 (X G) = (X ∩ ∼ G)
7520clos1base 5878 . . . . . . . . . . . . . 14 (X ran R) G
76 sscon34 3661 . . . . . . . . . . . . . 14 ((X ran R) G ↔ ∼ G ∼ (X ran R))
7775, 76mpbi 199 . . . . . . . . . . . . 13 G ∼ (X ran R)
78 df-dif 3215 . . . . . . . . . . . . . . 15 (X ran R) = (X ∩ ∼ ran R)
7978compleqi 3244 . . . . . . . . . . . . . 14 ∼ (X ran R) = ∼ (X ∩ ∼ ran R)
80 iinun 3548 . . . . . . . . . . . . . 14 ∼ (X ∩ ∼ ran R) = ( ∼ X ∪ ∼ ∼ ran R)
81 dblcompl 3227 . . . . . . . . . . . . . . 15 ∼ ∼ ran R = ran R
8281uneq2i 3415 . . . . . . . . . . . . . 14 ( ∼ X ∪ ∼ ∼ ran R) = ( ∼ X ∪ ran R)
8379, 80, 823eqtri 2377 . . . . . . . . . . . . 13 ∼ (X ran R) = ( ∼ X ∪ ran R)
8477, 83sseqtri 3303 . . . . . . . . . . . 12 G ( ∼ X ∪ ran R)
85 sslin 3481 . . . . . . . . . . . 12 ( ∼ G ( ∼ X ∪ ran R) → (X ∩ ∼ G) (X ∩ ( ∼ X ∪ ran R)))
8684, 85ax-mp 5 . . . . . . . . . . 11 (X ∩ ∼ G) (X ∩ ( ∼ X ∪ ran R))
8774, 86eqsstri 3301 . . . . . . . . . 10 (X G) (X ∩ ( ∼ X ∪ ran R))
88 indi 3501 . . . . . . . . . . . 12 (X ∩ ( ∼ X ∪ ran R)) = ((X ∩ ∼ X) ∪ (X ∩ ran R))
89 incompl 4073 . . . . . . . . . . . . 13 (X ∩ ∼ X) =
9089uneq1i 3414 . . . . . . . . . . . 12 ((X ∩ ∼ X) ∪ (X ∩ ran R)) = ( ∪ (X ∩ ran R))
91 uncom 3408 . . . . . . . . . . . . 13 ( ∪ (X ∩ ran R)) = ((X ∩ ran R) ∪ )
92 un0 3575 . . . . . . . . . . . . 13 ((X ∩ ran R) ∪ ) = (X ∩ ran R)
9391, 92eqtri 2373 . . . . . . . . . . . 12 ( ∪ (X ∩ ran R)) = (X ∩ ran R)
9488, 90, 933eqtri 2377 . . . . . . . . . . 11 (X ∩ ( ∼ X ∪ ran R)) = (X ∩ ran R)
95 inss2 3476 . . . . . . . . . . 11 (X ∩ ran R) ran R
9694, 95eqsstri 3301 . . . . . . . . . 10 (X ∩ ( ∼ X ∪ ran R)) ran R
9787, 96sstri 3281 . . . . . . . . 9 (X G) ran R
9873, 97jctil 523 . . . . . . . 8 (((Fun R Fun R) (X dom R ran R X)) → ((X G) ran R (X G) G))
99 ssin 3477 . . . . . . . 8 (((X G) ran R (X G) G) ↔ (X G) (ran R ∩ ∼ G))
10098, 99sylib 188 . . . . . . 7 (((Fun R Fun R) (X dom R ran R X)) → (X G) (ran R ∩ ∼ G))
101 sbthlem1.7 . . . . . . . 8 D = (ran R G)
102 df-dif 3215 . . . . . . . 8 (ran R G) = (ran R ∩ ∼ G)
103101, 102eqtri 2373 . . . . . . 7 D = (ran R ∩ ∼ G)
104100, 68, 1033sstr4g 3312 . . . . . 6 (((Fun R Fun R) (X dom R ran R X)) → B D)
105 ssdif 3401 . . . . . . . 8 (ran R X → (ran R G) (X G))
10645, 105syl 15 . . . . . . 7 (((Fun R Fun R) (X dom R ran R X)) → (ran R G) (X G))
107106, 101, 683sstr4g 3312 . . . . . 6 (((Fun R Fun R) (X dom R ran R X)) → D B)
108104, 107eqssd 3289 . . . . 5 (((Fun R Fun R) (X dom R ran R X)) → B = D)
10971, 108syl5breq 4674 . . . 4 (((Fun R Fun R) (X dom R ran R X)) → BD)
11010, 68ineq12i 3455 . . . . . 6 (AB) = ((XG) ∩ (X G))
111 inindif 4075 . . . . . 6 ((XG) ∩ (X G)) =
112110, 111eqtri 2373 . . . . 5 (AB) =
11330, 101ineq12i 3455 . . . . . 6 (CD) = ((ran RG) ∩ (ran R G))
114 inindif 4075 . . . . . 6 ((ran RG) ∩ (ran R G)) =
115113, 114eqtri 2373 . . . . 5 (CD) =
116 unen 6048 . . . . 5 (((AC BD) ((AB) = (CD) = )) → (AB) ≈ (CD))
117112, 115, 116mpanr12 666 . . . 4 ((AC BD) → (AB) ≈ (CD))
11867, 109, 117syl2anc 642 . . 3 (((Fun R Fun R) (X dom R ran R X)) → (AB) ≈ (CD))
119 ensym 6037 . . 3 ((AB) ≈ (CD) ↔ (CD) ≈ (AB))
120118, 119sylib 188 . 2 (((Fun R Fun R) (X dom R ran R X)) → (CD) ≈ (AB))
12130, 101uneq12i 3416 . . 3 (CD) = ((ran RG) ∪ (ran R G))
122 inundif 3628 . . 3 ((ran RG) ∪ (ran R G)) = ran R
123121, 122eqtri 2373 . 2 (CD) = ran R
12410, 68uneq12i 3416 . . 3 (AB) = ((XG) ∪ (X G))
125 inundif 3628 . . 3 ((XG) ∪ (X G)) = X
126124, 125eqtri 2373 . 2 (AB) = X
127120, 123, 1263brtr3g 4670 1 (((Fun R Fun R) (X dom R ran R X)) → ran RX)
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   = wceq 1642   wcel 1710  Vcvv 2859  ccompl 3205   cdif 3206  cun 3207  cin 3208   wss 3257  c0 3550   class class class wbr 4639  cima 4722  ccnv 4771  dom cdm 4772  ran crn 4773   cres 4774  Fun wfun 4775   Fn wfn 4776  –→wf 4777  1-1wf1 4778  1-1-ontowf1o 4780   Clos1 cclos1 5872  cen 6028
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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  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-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873  df-en 6029
This theorem is referenced by:  sbthlem2  6204
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