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Theorem fnasrn 5417
 Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
fnasrn (F Fn AF = ran {x, y (x A y = x, (Fx))})
Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fnasrn
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fndm 5182 . . . . . 6 (F Fn A → dom F = A)
2 opeldm 4910 . . . . . . 7 (z, w Fz dom F)
3 eleq2 2414 . . . . . . 7 (dom F = A → (z dom Fz A))
42, 3syl5ib 210 . . . . . 6 (dom F = A → (z, w Fz A))
51, 4syl 15 . . . . 5 (F Fn A → (z, w Fz A))
6 eleq1 2413 . . . . . . . . 9 (x = z → (x Az A))
76biimpcd 215 . . . . . . . 8 (x A → (x = zz A))
87adantrd 454 . . . . . . 7 (x A → ((x = z (Fx) = w) → z A))
98rexlimiv 2732 . . . . . 6 (x A (x = z (Fx) = w) → z A)
109a1i 10 . . . . 5 (F Fn A → (x A (x = z (Fx) = w) → z A))
11 fveq2 5328 . . . . . . . . . 10 (x = z → (Fx) = (Fz))
1211eqeq1d 2361 . . . . . . . . 9 (x = z → ((Fx) = w ↔ (Fz) = w))
1312ceqsrexv 2972 . . . . . . . 8 (z A → (x A (x = z (Fx) = w) ↔ (Fz) = w))
1413adantl 452 . . . . . . 7 ((F Fn A z A) → (x A (x = z (Fx) = w) ↔ (Fz) = w))
15 fnopfvb 5359 . . . . . . 7 ((F Fn A z A) → ((Fz) = wz, w F))
1614, 15bitr2d 245 . . . . . 6 ((F Fn A z A) → (z, w Fx A (x = z (Fx) = w)))
1716ex 423 . . . . 5 (F Fn A → (z A → (z, w Fx A (x = z (Fx) = w))))
185, 10, 17pm5.21ndd 343 . . . 4 (F Fn A → (z, w Fx A (x = z (Fx) = w)))
19 vex 2862 . . . . . 6 z V
20 vex 2862 . . . . . 6 w V
2119, 20opex 4588 . . . . 5 z, w V
22 eqeq1 2359 . . . . . . 7 (y = z, w → (y = x, (Fx)z, w = x, (Fx)))
23 eqcom 2355 . . . . . . . 8 (z, w = x, (Fx)x, (Fx) = z, w)
24 opth 4602 . . . . . . . 8 (x, (Fx) = z, w ↔ (x = z (Fx) = w))
2523, 24bitri 240 . . . . . . 7 (z, w = x, (Fx) ↔ (x = z (Fx) = w))
2622, 25syl6bb 252 . . . . . 6 (y = z, w → (y = x, (Fx) ↔ (x = z (Fx) = w)))
2726rexbidv 2635 . . . . 5 (y = z, w → (x A y = x, (Fx)x A (x = z (Fx) = w)))
2821, 27elab 2985 . . . 4 (z, w {y x A y = x, (Fx)} ↔ x A (x = z (Fx) = w))
2918, 28syl6bbr 254 . . 3 (F Fn A → (z, w Fz, w {y x A y = x, (Fx)}))
3029eqrelrdv 4852 . 2 (F Fn AF = {y x A y = x, (Fx)})
31 rnopab2 4968 . 2 ran {x, y (x A y = x, (Fx))} = {y x A y = x, (Fx)}
3230, 31syl6eqr 2403 1 (F Fn AF = ran {x, y (x A y = x, (Fx))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  ⟨cop 4561  {copab 4622  dom cdm 4772  ran crn 4773   Fn wfn 4776   ‘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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by: (None)
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