Theorem f1fveq 5474

Description: Equality of function values for a one-to-one function. (Contributed by set.mm contributors, 11-Feb-1997.)

Assertion

Ref	Expression
f1fveq	⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → ((F ‘C) = (F ‘D) ↔ C = D))

Proof of Theorem f1fveq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5329	. . . . . . 7 ⊢ (x = C → (F ‘x) = (F ‘C))
2	1	eqeq1d 2361	. . . . . 6 ⊢ (x = C → ((F ‘x) = (F ‘y) ↔ (F ‘C) = (F ‘y)))
3		eqeq1 2359	. . . . . 6 ⊢ (x = C → (x = y ↔ C = y))
4	2, 3	imbi12d 311	. . . . 5 ⊢ (x = C → (((F ‘x) = (F ‘y) → x = y) ↔ ((F ‘C) = (F ‘y) → C = y)))
5	4	imbi2d 307	. . . 4 ⊢ (x = C → ((F:A–1-1→B → ((F ‘x) = (F ‘y) → x = y)) ↔ (F:A–1-1→B → ((F ‘C) = (F ‘y) → C = y))))
6		fveq2 5329	. . . . . . 7 ⊢ (y = D → (F ‘y) = (F ‘D))
7	6	eqeq2d 2364	. . . . . 6 ⊢ (y = D → ((F ‘C) = (F ‘y) ↔ (F ‘C) = (F ‘D)))
8		eqeq2 2362	. . . . . 6 ⊢ (y = D → (C = y ↔ C = D))
9	7, 8	imbi12d 311	. . . . 5 ⊢ (y = D → (((F ‘C) = (F ‘y) → C = y) ↔ ((F ‘C) = (F ‘D) → C = D)))
10	9	imbi2d 307	. . . 4 ⊢ (y = D → ((F:A–1-1→B → ((F ‘C) = (F ‘y) → C = y)) ↔ (F:A–1-1→B → ((F ‘C) = (F ‘D) → C = D))))
11		dff13 5472	. . . . . . 7 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
12	11	simprbi 450	. . . . . 6 ⊢ (F:A–1-1→B → ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
13		rsp2 2677	. . . . . 6 ⊢ (∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y) → ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
14	12, 13	syl 15	. . . . 5 ⊢ (F:A–1-1→B → ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
15	14	com12 27	. . . 4 ⊢ ((x ∈ A ∧ y ∈ A) → (F:A–1-1→B → ((F ‘x) = (F ‘y) → x = y)))
16	5, 10, 15	vtocl2ga 2923	. . 3 ⊢ ((C ∈ A ∧ D ∈ A) → (F:A–1-1→B → ((F ‘C) = (F ‘D) → C = D)))
17	16	impcom 419	. 2 ⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → ((F ‘C) = (F ‘D) → C = D))
18		fveq2 5329	. 2 ⊢ (C = D → (F ‘C) = (F ‘D))
19	17, 18	impbid1 194	1 ⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → ((F ‘C) = (F ‘D) ↔ C = D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  –→wf 4778  –1-1→wf1 4779   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fv 4796

This theorem is referenced by:  f1elima  5475  f1oiso  5500