Theorem fconst5 5456

Description: Two ways to express that a function is constant. (Contributed by set.mm contributors, 27-Nov-2007.)

Assertion

Ref	Expression
fconst5	⊢ ((F Fn A ∧ A ≠ ∅) → (F = (A × {B}) ↔ ran F = {B}))

Proof of Theorem fconst5

Step	Hyp	Ref	Expression
1		rneq 4957	. . . 4 ⊢ (F = (A × {B}) → ran F = ran (A × {B}))
2		rnxp 5052	. . . . 5 ⊢ (A ≠ ∅ → ran (A × {B}) = {B})
3	2	eqeq2d 2364	. . . 4 ⊢ (A ≠ ∅ → (ran F = ran (A × {B}) ↔ ran F = {B}))
4	1, 3	syl5ib 210	. . 3 ⊢ (A ≠ ∅ → (F = (A × {B}) → ran F = {B}))
5	4	adantl 452	. 2 ⊢ ((F Fn A ∧ A ≠ ∅) → (F = (A × {B}) → ran F = {B}))
6		df-fo 4794	. . . . . . 7 ⊢ (F:A–onto→{B} ↔ (F Fn A ∧ ran F = {B}))
7		fof 5270	. . . . . . 7 ⊢ (F:A–onto→{B} → F:A–→{B})
8	6, 7	sylbir 204	. . . . . 6 ⊢ ((F Fn A ∧ ran F = {B}) → F:A–→{B})
9		fconst2g 5453	. . . . . 6 ⊢ (B ∈ V → (F:A–→{B} ↔ F = (A × {B})))
10	8, 9	syl5ib 210	. . . . 5 ⊢ (B ∈ V → ((F Fn A ∧ ran F = {B}) → F = (A × {B})))
11	10	exp3a 425	. . . 4 ⊢ (B ∈ V → (F Fn A → (ran F = {B} → F = (A × {B}))))
12	11	adantrd 454	. . 3 ⊢ (B ∈ V → ((F Fn A ∧ A ≠ ∅) → (ran F = {B} → F = (A × {B}))))
13		rneq0 4971	. . . . . . 7 ⊢ (F = ∅ ↔ ran F = ∅)
14	13	a1i 10	. . . . . 6 ⊢ (¬ B ∈ V → (F = ∅ ↔ ran F = ∅))
15		snprc 3789	. . . . . . . . . 10 ⊢ (¬ B ∈ V ↔ {B} = ∅)
16	15	biimpi 186	. . . . . . . . 9 ⊢ (¬ B ∈ V → {B} = ∅)
17	16	xpeq2d 4809	. . . . . . . 8 ⊢ (¬ B ∈ V → (A × {B}) = (A × ∅))
18		xp0 5045	. . . . . . . 8 ⊢ (A × ∅) = ∅
19	17, 18	syl6eq 2401	. . . . . . 7 ⊢ (¬ B ∈ V → (A × {B}) = ∅)
20	19	eqeq2d 2364	. . . . . 6 ⊢ (¬ B ∈ V → (F = (A × {B}) ↔ F = ∅))
21	16	eqeq2d 2364	. . . . . 6 ⊢ (¬ B ∈ V → (ran F = {B} ↔ ran F = ∅))
22	14, 20, 21	3bitr4d 276	. . . . 5 ⊢ (¬ B ∈ V → (F = (A × {B}) ↔ ran F = {B}))
23	22	biimprd 214	. . . 4 ⊢ (¬ B ∈ V → (ran F = {B} → F = (A × {B})))
24	23	a1d 22	. . 3 ⊢ (¬ B ∈ V → ((F Fn A ∧ A ≠ ∅) → (ran F = {B} → F = (A × {B}))))
25	12, 24	pm2.61i 156	. 2 ⊢ ((F Fn A ∧ A ≠ ∅) → (ran F = {B} → F = (A × {B})))
26	5, 25	impbid 183	1 ⊢ ((F Fn A ∧ A ≠ ∅) → (F = (A × {B}) ↔ ran F = {B}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860  ∅c0 3551  {csn 3738   × cxp 4771  ran crn 4774   Fn wfn 4777  –→wf 4778  –onto→wfo 4780

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-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796

This theorem is referenced by: (None)