Theorem dff13f 5472

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Hypotheses

Ref	Expression
dff13f.1	⊢ ℲxF
dff13f.2	⊢ ℲyF

Assertion

Ref	Expression
dff13f	⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Distinct variable group:   x,y,A

Allowed substitution hints:   B(x,y)   F(x,y)

Proof of Theorem dff13f

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff13 5471	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v)))
2		dff13f.2	. . . . . . . . 9 ⊢ ℲyF
3		nfcv 2489	. . . . . . . . 9 ⊢ Ⅎyw
4	2, 3	nffv 5334	. . . . . . . 8 ⊢ Ⅎy(F ‘w)
5		nfcv 2489	. . . . . . . . 9 ⊢ Ⅎyv
6	2, 5	nffv 5334	. . . . . . . 8 ⊢ Ⅎy(F ‘v)
7	4, 6	nfeq 2496	. . . . . . 7 ⊢ Ⅎy(F ‘w) = (F ‘v)
8		nfv 1619	. . . . . . 7 ⊢ Ⅎy w = v
9	7, 8	nfim 1813	. . . . . 6 ⊢ Ⅎy((F ‘w) = (F ‘v) → w = v)
10		nfv 1619	. . . . . 6 ⊢ Ⅎv((F ‘w) = (F ‘y) → w = y)
11		fveq2 5328	. . . . . . . 8 ⊢ (v = y → (F ‘v) = (F ‘y))
12	11	eqeq2d 2364	. . . . . . 7 ⊢ (v = y → ((F ‘w) = (F ‘v) ↔ (F ‘w) = (F ‘y)))
13		eqeq2 2362	. . . . . . 7 ⊢ (v = y → (w = v ↔ w = y))
14	12, 13	imbi12d 311	. . . . . 6 ⊢ (v = y → (((F ‘w) = (F ‘v) → w = v) ↔ ((F ‘w) = (F ‘y) → w = y)))
15	9, 10, 14	cbvral 2831	. . . . 5 ⊢ (∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
16	15	ralbii 2638	. . . 4 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀w ∈ A ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
17		nfcv 2489	. . . . . 6 ⊢ ℲxA
18		dff13f.1	. . . . . . . . 9 ⊢ ℲxF
19		nfcv 2489	. . . . . . . . 9 ⊢ Ⅎxw
20	18, 19	nffv 5334	. . . . . . . 8 ⊢ Ⅎx(F ‘w)
21		nfcv 2489	. . . . . . . . 9 ⊢ Ⅎxy
22	18, 21	nffv 5334	. . . . . . . 8 ⊢ Ⅎx(F ‘y)
23	20, 22	nfeq 2496	. . . . . . 7 ⊢ Ⅎx(F ‘w) = (F ‘y)
24		nfv 1619	. . . . . . 7 ⊢ Ⅎx w = y
25	23, 24	nfim 1813	. . . . . 6 ⊢ Ⅎx((F ‘w) = (F ‘y) → w = y)
26	17, 25	nfral 2667	. . . . 5 ⊢ Ⅎx∀y ∈ A ((F ‘w) = (F ‘y) → w = y)
27		nfv 1619	. . . . 5 ⊢ Ⅎw∀y ∈ A ((F ‘x) = (F ‘y) → x = y)
28		fveq2 5328	. . . . . . . 8 ⊢ (w = x → (F ‘w) = (F ‘x))
29	28	eqeq1d 2361	. . . . . . 7 ⊢ (w = x → ((F ‘w) = (F ‘y) ↔ (F ‘x) = (F ‘y)))
30		eqeq1 2359	. . . . . . 7 ⊢ (w = x → (w = y ↔ x = y))
31	29, 30	imbi12d 311	. . . . . 6 ⊢ (w = x → (((F ‘w) = (F ‘y) → w = y) ↔ ((F ‘x) = (F ‘y) → x = y)))
32	31	ralbidv 2634	. . . . 5 ⊢ (w = x → (∀y ∈ A ((F ‘w) = (F ‘y) → w = y) ↔ ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
33	26, 27, 32	cbvral 2831	. . . 4 ⊢ (∀w ∈ A ∀y ∈ A ((F ‘w) = (F ‘y) → w = y) ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
34	16, 33	bitri 240	. . 3 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
35	34	anbi2i 675	. 2 ⊢ ((F:A–→B ∧ ∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v)) ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
36	1, 35	bitri 240	1 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  Ⅎwnfc 2476  ∀wral 2614  –→wf 4777  –1-1→wf1 4778   ‘cfv 4781

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-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: (None)