Theorem f1fveq 2918

Description: Equality of function values for a one-to-one function.

Assertion

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

Proof of Theorem f1fveq

Step	Hyp	Ref	Expression
1		fveq2 2832	. . . . . . . 8 ⊢ (x = C → (F ‘x) = (F ‘C))
2	1	cleq1d 1109	. . . . . . 7 ⊢ (x = C → ((F ‘x) = (F ‘y) ↔ (F ‘C) = (F ‘y)))
3		cleq1 1107	. . . . . . 7 ⊢ (x = C → (x = y ↔ C = y))
4	2, 3	imbi12d 474	. . . . . 6 ⊢ (x = C → (((F ‘x) = (F ‘y) → x = y) ↔ ((F ‘C) = (F ‘y) → C = y)))
5	4	imbi2d 464	. . . . 5 ⊢ (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 2832	. . . . . . . 8 ⊢ (y = D → (F ‘y) = (F ‘D))
7	6	cleq2d 1112	. . . . . . 7 ⊢ (y = D → ((F ‘C) = (F ‘y) ↔ (F ‘C) = (F ‘D)))
8		cleq2 1110	. . . . . . 7 ⊢ (y = D → (C = y ↔ C = D))
9	7, 8	imbi12d 474	. . . . . 6 ⊢ (y = D → (((F ‘C) = (F ‘y) → C = y) ↔ ((F ‘C) = (F ‘D) → C = D)))
10	9	imbi2d 464	. . . . 5 ⊢ (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		f1fv 2916	. . . . . . . 8 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
12	11	pm3.27bd 263	. . . . . . 7 ⊢ (F:A–1-1→B → ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
13		ra42 1245	. . . . . . 7 ⊢ (∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y) → ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
14	12, 13	syl 12	. . . . . 6 ⊢ (F:A–1-1→B → ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
15	14	com12 13	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ A) → (F:A–1-1→B → ((F ‘x) = (F ‘y) → x = y)))
16	5, 10, 15	vtocl2ga 1388	. . . 4 ⊢ ((C ∈ A ∧ D ∈ A) → (F:A–1-1→B → ((F ‘C) = (F ‘D) → C = D)))
17	16	com12 13	. . 3 ⊢ (F:A–1-1→B → ((C ∈ A ∧ D ∈ A) → ((F ‘C) = (F ‘D) → C = D)))
18	17	imp 277	. 2 ⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → ((F ‘C) = (F ‘D) → C = D))
19		fveq2 2832	. . 3 ⊢ (C = D → (F ‘C) = (F ‘D))
20	19	a1i 7	. 2 ⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → (C = D → (F ‘C) = (F ‘D)))
21	18, 20	impbid 397	1 ⊢ ((F:A–1-1→B ∧ (C ∈ A ∧ D ∈ A)) → ((F ‘C) = (F ‘D) ↔ C = D))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  –→wf 2418  –1-1→wf1 2419   ‘cfv 2422

This theorem is referenced by:  isowe 2941  f1oiso 2942  f1oweOLD 2944  2dom 3332  xpdom2 3345  mapenlem2 3385

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438

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