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Theorem fvelimab 5370
 Description: Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.) (Contributed by set.mm contributors, 20-Jan-2007.) (Revised by set.mm contributors, 25-Dec-2011.)
Assertion
Ref Expression
fvelimab ((F Fn A B A) → (C (FB) ↔ x B (Fx) = C))
Distinct variable groups:   x,B   x,C   x,F
Allowed substitution hint:   A(x)

Proof of Theorem fvelimab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elex 2867 . . 3 (C (FB) → C V)
21anim2i 552 . 2 (((F Fn A B A) C (FB)) → ((F Fn A B A) C V))
3 fvex 5339 . . . . 5 (Fx) V
4 eleq1 2413 . . . . 5 ((Fx) = C → ((Fx) V ↔ C V))
53, 4mpbii 202 . . . 4 ((Fx) = CC V)
65rexlimivw 2734 . . 3 (x B (Fx) = CC V)
76anim2i 552 . 2 (((F Fn A B A) x B (Fx) = C) → ((F Fn A B A) C V))
8 eleq1 2413 . . . . . 6 (y = C → (y (FB) ↔ C (FB)))
9 eqeq2 2362 . . . . . . 7 (y = C → ((Fx) = y ↔ (Fx) = C))
109rexbidv 2635 . . . . . 6 (y = C → (x B (Fx) = yx B (Fx) = C))
118, 10bibi12d 312 . . . . 5 (y = C → ((y (FB) ↔ x B (Fx) = y) ↔ (C (FB) ↔ x B (Fx) = C)))
1211imbi2d 307 . . . 4 (y = C → (((F Fn A B A) → (y (FB) ↔ x B (Fx) = y)) ↔ ((F Fn A B A) → (C (FB) ↔ x B (Fx) = C))))
13 fnfun 5181 . . . . . . 7 (F Fn A → Fun F)
1413adantr 451 . . . . . 6 ((F Fn A B A) → Fun F)
15 fndm 5182 . . . . . . . 8 (F Fn A → dom F = A)
1615sseq2d 3299 . . . . . . 7 (F Fn A → (B dom FB A))
1716biimpar 471 . . . . . 6 ((F Fn A B A) → B dom F)
18 dfimafn 5366 . . . . . 6 ((Fun F B dom F) → (FB) = {y x B (Fx) = y})
1914, 17, 18syl2anc 642 . . . . 5 ((F Fn A B A) → (FB) = {y x B (Fx) = y})
2019abeq2d 2462 . . . 4 ((F Fn A B A) → (y (FB) ↔ x B (Fx) = y))
2112, 20vtoclg 2914 . . 3 (C V → ((F Fn A B A) → (C (FB) ↔ x B (Fx) = C)))
2221impcom 419 . 2 (((F Fn A B A) C V) → (C (FB) ↔ x B (Fx) = C))
232, 7, 22pm5.21nd 868 1 ((F Fn A B A) → (C (FB) ↔ x B (Fx) = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257   “ cima 4722  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781 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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  f1elima  5474  ovelimab  5610  dfnnc3  5885
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