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Theorem enmap2lem5 6067
 Description: Lemma for enmap2 6068. Calculate the range of W. (Contributed by SF, 26-Feb-2015.)
Hypothesis
Ref Expression
enmap2lem5.1 W = (s (Gm a) (s r))
Assertion
Ref Expression
enmap2lem5 (r:a1-1-ontob → ran W = (Gm b))
Distinct variable groups:   s,a   G,s   s,r
Allowed substitution hints:   G(r,a,b)   W(s,r,a,b)

Proof of Theorem enmap2lem5
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 enmap2lem5.1 . . . 4 W = (s (Gm a) (s r))
21enmap2lem2 6064 . . 3 W Fn (Gm a)
3 coeq1 4874 . . . . . . 7 (s = p → (s r) = (p r))
4 vex 2862 . . . . . . . 8 p V
5 vex 2862 . . . . . . . . 9 r V
65cnvex 5102 . . . . . . . 8 r V
74, 6coex 4750 . . . . . . 7 (p r) V
83, 1, 7fvmpt 5700 . . . . . 6 (p (Gm a) → (Wp) = (p r))
98adantl 452 . . . . 5 ((r:a1-1-ontob p (Gm a)) → (Wp) = (p r))
10 elmapi 6016 . . . . . . 7 (p (Gm a) → p:a–→G)
11 f1ocnv 5299 . . . . . . . 8 (r:a1-1-ontobr:b1-1-ontoa)
12 f1of 5287 . . . . . . . 8 (r:b1-1-ontoar:b–→a)
1311, 12syl 15 . . . . . . 7 (r:a1-1-ontobr:b–→a)
14 fco 5231 . . . . . . 7 ((p:a–→G r:b–→a) → (p r):b–→G)
1510, 13, 14syl2anr 464 . . . . . 6 ((r:a1-1-ontob p (Gm a)) → (p r):b–→G)
16 elovex1 5649 . . . . . . . 8 (p (Gm a) → G V)
17 vex 2862 . . . . . . . . 9 b V
18 elmapg 6012 . . . . . . . . 9 ((G V b V (p r) V) → ((p r) (Gm b) ↔ (p r):b–→G))
1917, 7, 18mp3an23 1269 . . . . . . . 8 (G V → ((p r) (Gm b) ↔ (p r):b–→G))
2016, 19syl 15 . . . . . . 7 (p (Gm a) → ((p r) (Gm b) ↔ (p r):b–→G))
2120adantl 452 . . . . . 6 ((r:a1-1-ontob p (Gm a)) → ((p r) (Gm b) ↔ (p r):b–→G))
2215, 21mpbird 223 . . . . 5 ((r:a1-1-ontob p (Gm a)) → (p r) (Gm b))
239, 22eqeltrd 2427 . . . 4 ((r:a1-1-ontob p (Gm a)) → (Wp) (Gm b))
2423ralrimiva 2697 . . 3 (r:a1-1-ontobp (Gm a)(Wp) (Gm b))
25 fnfvrnss 5429 . . 3 ((W Fn (Gm a) p (Gm a)(Wp) (Gm b)) → ran W (Gm b))
262, 24, 25sylancr 644 . 2 (r:a1-1-ontob → ran W (Gm b))
27 elmapi 6016 . . . . . . . . . 10 (p (Gm b) → p:b–→G)
28 f1of 5287 . . . . . . . . . 10 (r:a1-1-ontobr:a–→b)
29 fco 5231 . . . . . . . . . 10 ((p:b–→G r:a–→b) → (p r):a–→G)
3027, 28, 29syl2anr 464 . . . . . . . . 9 ((r:a1-1-ontob p (Gm b)) → (p r):a–→G)
31 elovex1 5649 . . . . . . . . . . 11 (p (Gm b) → G V)
32 vex 2862 . . . . . . . . . . . 12 a V
334, 5coex 4750 . . . . . . . . . . . 12 (p r) V
34 elmapg 6012 . . . . . . . . . . . 12 ((G V a V (p r) V) → ((p r) (Gm a) ↔ (p r):a–→G))
3532, 33, 34mp3an23 1269 . . . . . . . . . . 11 (G V → ((p r) (Gm a) ↔ (p r):a–→G))
3631, 35syl 15 . . . . . . . . . 10 (p (Gm b) → ((p r) (Gm a) ↔ (p r):a–→G))
3736adantl 452 . . . . . . . . 9 ((r:a1-1-ontob p (Gm b)) → ((p r) (Gm a) ↔ (p r):a–→G))
3830, 37mpbird 223 . . . . . . . 8 ((r:a1-1-ontob p (Gm b)) → (p r) (Gm a))
39 coeq1 4874 . . . . . . . . . 10 (s = (p r) → (s r) = ((p r) r))
40 coass 5097 . . . . . . . . . 10 ((p r) r) = (p (r r))
4139, 40syl6eq 2401 . . . . . . . . 9 (s = (p r) → (s r) = (p (r r)))
425, 6coex 4750 . . . . . . . . . 10 (r r) V
434, 42coex 4750 . . . . . . . . 9 (p (r r)) V
4441, 1, 43fvmpt 5700 . . . . . . . 8 ((p r) (Gm a) → (W ‘(p r)) = (p (r r)))
4538, 44syl 15 . . . . . . 7 ((r:a1-1-ontob p (Gm b)) → (W ‘(p r)) = (p (r r)))
46 f1ococnv2 5309 . . . . . . . . 9 (r:a1-1-ontob → (r r) = ( I b))
4746coeq2d 4879 . . . . . . . 8 (r:a1-1-ontob → (p (r r)) = (p ( I b)))
48 fcoi1 5240 . . . . . . . . 9 (p:b–→G → (p ( I b)) = p)
4927, 48syl 15 . . . . . . . 8 (p (Gm b) → (p ( I b)) = p)
5047, 49sylan9eq 2405 . . . . . . 7 ((r:a1-1-ontob p (Gm b)) → (p (r r)) = p)
5145, 50eqtrd 2385 . . . . . 6 ((r:a1-1-ontob p (Gm b)) → (W ‘(p r)) = p)
52 fnbrfvb 5358 . . . . . . 7 ((W Fn (Gm a) (p r) (Gm a)) → ((W ‘(p r)) = p ↔ (p r)Wp))
532, 38, 52sylancr 644 . . . . . 6 ((r:a1-1-ontob p (Gm b)) → ((W ‘(p r)) = p ↔ (p r)Wp))
5451, 53mpbid 201 . . . . 5 ((r:a1-1-ontob p (Gm b)) → (p r)Wp)
55 brelrn 4960 . . . . 5 ((p r)Wpp ran W)
5654, 55syl 15 . . . 4 ((r:a1-1-ontob p (Gm b)) → p ran W)
5756ex 423 . . 3 (r:a1-1-ontob → (p (Gm b) → p ran W))
5857ssrdv 3278 . 2 (r:a1-1-ontob → (Gm b) ran W)
5926, 58eqssd 3289 1 (r:a1-1-ontob → ran W = (Gm b))
 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  ran crn 4773   ↾ cres 4774   Fn wfn 4776  –→wf 4777  –1-1-onto→wf1o 4780   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑m cmap 5999 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-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:  enmap2  6068
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