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Theorem foco2 5426
Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
foco2 ((F:B–→C G:A–→B (F G):AontoC) → F:BontoC)

Proof of Theorem foco2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . 2 ((F:B–→C G:A–→B (F G):AontoC) → F:B–→C)
2 foelrn 5425 . . . . . 6 (((F G):AontoC y C) → z A y = ((F G) ‘z))
3 ffvelrn 5415 . . . . . . . . . 10 ((G:A–→B z A) → (Gz) B)
43adantll 694 . . . . . . . . 9 (((F:B–→C G:A–→B) z A) → (Gz) B)
5 fvco3 5384 . . . . . . . . . 10 ((G:A–→B z A) → ((F G) ‘z) = (F ‘(Gz)))
65adantll 694 . . . . . . . . 9 (((F:B–→C G:A–→B) z A) → ((F G) ‘z) = (F ‘(Gz)))
7 fveq2 5328 . . . . . . . . . . 11 (x = (Gz) → (Fx) = (F ‘(Gz)))
87eqeq2d 2364 . . . . . . . . . 10 (x = (Gz) → (((F G) ‘z) = (Fx) ↔ ((F G) ‘z) = (F ‘(Gz))))
98rspcev 2955 . . . . . . . . 9 (((Gz) B ((F G) ‘z) = (F ‘(Gz))) → x B ((F G) ‘z) = (Fx))
104, 6, 9syl2anc 642 . . . . . . . 8 (((F:B–→C G:A–→B) z A) → x B ((F G) ‘z) = (Fx))
11 eqeq1 2359 . . . . . . . . 9 (y = ((F G) ‘z) → (y = (Fx) ↔ ((F G) ‘z) = (Fx)))
1211rexbidv 2635 . . . . . . . 8 (y = ((F G) ‘z) → (x B y = (Fx) ↔ x B ((F G) ‘z) = (Fx)))
1310, 12syl5ibrcom 213 . . . . . . 7 (((F:B–→C G:A–→B) z A) → (y = ((F G) ‘z) → x B y = (Fx)))
1413rexlimdva 2738 . . . . . 6 ((F:B–→C G:A–→B) → (z A y = ((F G) ‘z) → x B y = (Fx)))
152, 14syl5 28 . . . . 5 ((F:B–→C G:A–→B) → (((F G):AontoC y C) → x B y = (Fx)))
1615impl 603 . . . 4 ((((F:B–→C G:A–→B) (F G):AontoC) y C) → x B y = (Fx))
1716ralrimiva 2697 . . 3 (((F:B–→C G:A–→B) (F G):AontoC) → y C x B y = (Fx))
18173impa 1146 . 2 ((F:B–→C G:A–→B (F G):AontoC) → y C x B y = (Fx))
19 dffo3 5422 . 2 (F:BontoC ↔ (F:B–→C y C x B y = (Fx)))
201, 18, 19sylanbrc 645 1 ((F:B–→C G:A–→B (F G):AontoC) → F:BontoC)
Colors of variables: wff setvar class
Syntax hints:  wi 4   wa 358   w3a 934   = wceq 1642   wcel 1710  wral 2614  wrex 2615   ccom 4721  –→wf 4777  ontowfo 4779  cfv 4781
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-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-fo 4793  df-fv 4795
This theorem is referenced by: (None)
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