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Theorem f1fveq 5473
 Description: Equality of function values for a one-to-one function. (Contributed by set.mm contributors, 11-Feb-1997.)
Assertion
Ref Expression
f1fveq ((F:A1-1B (C A D A)) → ((FC) = (FD) ↔ C = D))

Proof of Theorem f1fveq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5328 . . . . . . 7 (x = C → (Fx) = (FC))
21eqeq1d 2361 . . . . . 6 (x = C → ((Fx) = (Fy) ↔ (FC) = (Fy)))
3 eqeq1 2359 . . . . . 6 (x = C → (x = yC = y))
42, 3imbi12d 311 . . . . 5 (x = C → (((Fx) = (Fy) → x = y) ↔ ((FC) = (Fy) → C = y)))
54imbi2d 307 . . . 4 (x = C → ((F:A1-1B → ((Fx) = (Fy) → x = y)) ↔ (F:A1-1B → ((FC) = (Fy) → C = y))))
6 fveq2 5328 . . . . . . 7 (y = D → (Fy) = (FD))
76eqeq2d 2364 . . . . . 6 (y = D → ((FC) = (Fy) ↔ (FC) = (FD)))
8 eqeq2 2362 . . . . . 6 (y = D → (C = yC = D))
97, 8imbi12d 311 . . . . 5 (y = D → (((FC) = (Fy) → C = y) ↔ ((FC) = (FD) → C = D)))
109imbi2d 307 . . . 4 (y = D → ((F:A1-1B → ((FC) = (Fy) → C = y)) ↔ (F:A1-1B → ((FC) = (FD) → C = D))))
11 dff13 5471 . . . . . . 7 (F:A1-1B ↔ (F:A–→B x A y A ((Fx) = (Fy) → x = y)))
1211simprbi 450 . . . . . 6 (F:A1-1Bx A y A ((Fx) = (Fy) → x = y))
13 rsp2 2676 . . . . . 6 (x A y A ((Fx) = (Fy) → x = y) → ((x A y A) → ((Fx) = (Fy) → x = y)))
1412, 13syl 15 . . . . 5 (F:A1-1B → ((x A y A) → ((Fx) = (Fy) → x = y)))
1514com12 27 . . . 4 ((x A y A) → (F:A1-1B → ((Fx) = (Fy) → x = y)))
165, 10, 15vtocl2ga 2922 . . 3 ((C A D A) → (F:A1-1B → ((FC) = (FD) → C = D)))
1716impcom 419 . 2 ((F:A1-1B (C A D A)) → ((FC) = (FD) → C = D))
18 fveq2 5328 . 2 (C = D → (FC) = (FD))
1917, 18impbid1 194 1 ((F:A1-1B (C A D A)) → ((FC) = (FD) ↔ C = D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  –→wf 4777  –1-1→wf1 4778   ‘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-f 4791  df-f1 4792  df-fv 4795 This theorem is referenced by:  f1elima  5474  f1oiso  5499
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