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Theorem enmap1lem5 6073
 Description: Lemma for enmap2 6068. Calculate the range of W. (Contributed by SF, 3-Mar-2015.)
Hypothesis
Ref Expression
enmap1lem5.1 W = (s (Am G) (r s))
Assertion
Ref Expression
enmap1lem5 (r:A1-1-ontoB → ran W = (Bm G))
Distinct variable groups:   G,s   s,r   A,s
Allowed substitution hints:   A(r)   B(s,r)   G(r)   W(s,r)

Proof of Theorem enmap1lem5
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 enmap1lem5.1 . . . 4 W = (s (Am G) (r s))
21enmap1lem2 6070 . . 3 W Fn (Am G)
3 coeq2 4875 . . . . . . 7 (s = p → (r s) = (r p))
4 vex 2862 . . . . . . . 8 r V
5 vex 2862 . . . . . . . 8 p V
64, 5coex 4750 . . . . . . 7 (r p) V
73, 1, 6fvmpt 5700 . . . . . 6 (p (Am G) → (Wp) = (r p))
87adantl 452 . . . . 5 ((r:A1-1-ontoB p (Am G)) → (Wp) = (r p))
9 f1of 5287 . . . . . . 7 (r:A1-1-ontoBr:A–→B)
10 elmapi 6016 . . . . . . 7 (p (Am G) → p:G–→A)
11 fco 5231 . . . . . . 7 ((r:A–→B p:G–→A) → (r p):G–→B)
129, 10, 11syl2an 463 . . . . . 6 ((r:A1-1-ontoB p (Am G)) → (r p):G–→B)
13 f1ofo 5293 . . . . . . . . 9 (r:A1-1-ontoBr:AontoB)
14 forn 5272 . . . . . . . . 9 (r:AontoB → ran r = B)
1513, 14syl 15 . . . . . . . 8 (r:A1-1-ontoB → ran r = B)
164rnex 5107 . . . . . . . 8 ran r V
1715, 16syl6eqelr 2442 . . . . . . 7 (r:A1-1-ontoBB V)
18 elovex2 5650 . . . . . . 7 (p (Am G) → G V)
19 elmapg 6012 . . . . . . . 8 ((B V G V (r p) V) → ((r p) (Bm G) ↔ (r p):G–→B))
206, 19mp3an3 1266 . . . . . . 7 ((B V G V) → ((r p) (Bm G) ↔ (r p):G–→B))
2117, 18, 20syl2an 463 . . . . . 6 ((r:A1-1-ontoB p (Am G)) → ((r p) (Bm G) ↔ (r p):G–→B))
2212, 21mpbird 223 . . . . 5 ((r:A1-1-ontoB p (Am G)) → (r p) (Bm G))
238, 22eqeltrd 2427 . . . 4 ((r:A1-1-ontoB p (Am G)) → (Wp) (Bm G))
2423ralrimiva 2697 . . 3 (r:A1-1-ontoBp (Am G)(Wp) (Bm G))
25 fnfvrnss 5429 . . 3 ((W Fn (Am G) p (Am G)(Wp) (Bm G)) → ran W (Bm G))
262, 24, 25sylancr 644 . 2 (r:A1-1-ontoB → ran W (Bm G))
27 f1ocnv 5299 . . . . . . . . . . 11 (r:A1-1-ontoBr:B1-1-ontoA)
28 f1of 5287 . . . . . . . . . . 11 (r:B1-1-ontoAr:B–→A)
2927, 28syl 15 . . . . . . . . . 10 (r:A1-1-ontoBr:B–→A)
30 elmapi 6016 . . . . . . . . . 10 (p (Bm G) → p:G–→B)
31 fco 5231 . . . . . . . . . 10 ((r:B–→A p:G–→B) → (r p):G–→A)
3229, 30, 31syl2an 463 . . . . . . . . 9 ((r:A1-1-ontoB p (Bm G)) → (r p):G–→A)
33 f1odm 5290 . . . . . . . . . . 11 (r:A1-1-ontoB → dom r = A)
344dmex 5106 . . . . . . . . . . 11 dom r V
3533, 34syl6eqelr 2442 . . . . . . . . . 10 (r:A1-1-ontoBA V)
36 elovex2 5650 . . . . . . . . . 10 (p (Bm G) → G V)
374cnvex 5102 . . . . . . . . . . . 12 r V
3837, 5coex 4750 . . . . . . . . . . 11 (r p) V
39 elmapg 6012 . . . . . . . . . . 11 ((A V G V (r p) V) → ((r p) (Am G) ↔ (r p):G–→A))
4038, 39mp3an3 1266 . . . . . . . . . 10 ((A V G V) → ((r p) (Am G) ↔ (r p):G–→A))
4135, 36, 40syl2an 463 . . . . . . . . 9 ((r:A1-1-ontoB p (Bm G)) → ((r p) (Am G) ↔ (r p):G–→A))
4232, 41mpbird 223 . . . . . . . 8 ((r:A1-1-ontoB p (Bm G)) → (r p) (Am G))
43 coeq2 4875 . . . . . . . . 9 (s = (r p) → (r s) = (r (r p)))
444, 38coex 4750 . . . . . . . . 9 (r (r p)) V
4543, 1, 44fvmpt 5700 . . . . . . . 8 ((r p) (Am G) → (W ‘(r p)) = (r (r p)))
4642, 45syl 15 . . . . . . 7 ((r:A1-1-ontoB p (Bm G)) → (W ‘(r p)) = (r (r p)))
47 coass 5097 . . . . . . . 8 ((r r) p) = (r (r p))
48 f1ococnv2 5309 . . . . . . . . . 10 (r:A1-1-ontoB → (r r) = ( I B))
4948coeq1d 4878 . . . . . . . . 9 (r:A1-1-ontoB → ((r r) p) = (( I B) p))
50 fcoi2 5241 . . . . . . . . . 10 (p:G–→B → (( I B) p) = p)
5130, 50syl 15 . . . . . . . . 9 (p (Bm G) → (( I B) p) = p)
5249, 51sylan9eq 2405 . . . . . . . 8 ((r:A1-1-ontoB p (Bm G)) → ((r r) p) = p)
5347, 52syl5eqr 2399 . . . . . . 7 ((r:A1-1-ontoB p (Bm G)) → (r (r p)) = p)
5446, 53eqtrd 2385 . . . . . 6 ((r:A1-1-ontoB p (Bm G)) → (W ‘(r p)) = p)
55 fnbrfvb 5358 . . . . . . 7 ((W Fn (Am G) (r p) (Am G)) → ((W ‘(r p)) = p ↔ (r p)Wp))
562, 42, 55sylancr 644 . . . . . 6 ((r:A1-1-ontoB p (Bm G)) → ((W ‘(r p)) = p ↔ (r p)Wp))
5754, 56mpbid 201 . . . . 5 ((r:A1-1-ontoB p (Bm G)) → (r p)Wp)
58 brelrn 4960 . . . . 5 ((r p)Wpp ran W)
5957, 58syl 15 . . . 4 ((r:A1-1-ontoB p (Bm G)) → p ran W)
6059ex 423 . . 3 (r:A1-1-ontoB → (p (Bm G) → p ran W))
6160ssrdv 3278 . 2 (r:A1-1-ontoB → (Bm G) ran W)
6226, 61eqssd 3289 1 (r:A1-1-ontoB → ran W = (Bm G))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721   I cid 4763  ◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774   Fn wfn 4776  –→wf 4777  –onto→wfo 4779  –1-1-onto→wf1o 4780   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑m cmap 5999 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-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-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001 This theorem is referenced by:  enmap1  6074
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