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Theorem mptpreima 5682
 Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1 F = (x A B)
Assertion
Ref Expression
mptpreima (FC) = {x A B C}
Distinct variable group:   x,C
Allowed substitution hints:   A(x)   B(x)   F(x)

Proof of Theorem mptpreima
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dmmpt2.1 . . . . . 6 F = (x A B)
2 df-mpt 5652 . . . . . 6 (x A B) = {x, y (x A y = B)}
31, 2eqtri 2373 . . . . 5 F = {x, y (x A y = B)}
43cnveqi 4887 . . . 4 F = {x, y (x A y = B)}
5 cnvopab 5030 . . . 4 {x, y (x A y = B)} = {y, x (x A y = B)}
64, 5eqtri 2373 . . 3 F = {y, x (x A y = B)}
76imaeq1i 4939 . 2 (FC) = ({y, x (x A y = B)} “ C)
8 dfima3 4951 . . 3 ({y, x (x A y = B)} “ C) = ran ({y, x (x A y = B)} C)
9 resopab 4999 . . . . 5 ({y, x (x A y = B)} C) = {y, x (y C (x A y = B))}
109rneqi 4957 . . . 4 ran ({y, x (x A y = B)} C) = ran {y, x (y C (x A y = B))}
11 ancom 437 . . . . . . . . 9 ((y C (x A y = B)) ↔ ((x A y = B) y C))
12 anass 630 . . . . . . . . 9 (((x A y = B) y C) ↔ (x A (y = B y C)))
1311, 12bitri 240 . . . . . . . 8 ((y C (x A y = B)) ↔ (x A (y = B y C)))
1413exbii 1582 . . . . . . 7 (y(y C (x A y = B)) ↔ y(x A (y = B y C)))
15 19.42v 1905 . . . . . . . 8 (y(x A (y = B y C)) ↔ (x A y(y = B y C)))
16 df-clel 2349 . . . . . . . . . 10 (B Cy(y = B y C))
1716bicomi 193 . . . . . . . . 9 (y(y = B y C) ↔ B C)
1817anbi2i 675 . . . . . . . 8 ((x A y(y = B y C)) ↔ (x A B C))
1915, 18bitri 240 . . . . . . 7 (y(x A (y = B y C)) ↔ (x A B C))
2014, 19bitri 240 . . . . . 6 (y(y C (x A y = B)) ↔ (x A B C))
2120abbii 2465 . . . . 5 {x y(y C (x A y = B))} = {x (x A B C)}
22 rnopab 4967 . . . . 5 ran {y, x (y C (x A y = B))} = {x y(y C (x A y = B))}
23 df-rab 2623 . . . . 5 {x A B C} = {x (x A B C)}
2421, 22, 233eqtr4i 2383 . . . 4 ran {y, x (y C (x A y = B))} = {x A B C}
2510, 24eqtri 2373 . . 3 ran ({y, x (x A y = B)} C) = {x A B C}
268, 25eqtri 2373 . 2 ({y, x (x A y = B)} “ C) = {x A B C}
277, 26eqtri 2373 1 (FC) = {x A B C}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  {copab 4622   “ cima 4722  ◡ccnv 4771  ran crn 4773   ↾ cres 4774   ↦ cmpt 5651 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-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-res 4788  df-mpt 5652 This theorem is referenced by:  fmpt  5692
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