Theorem fconst5 5455

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 4956	. . . 4 ⊢ (F = (A × {B}) → ran F = ran (A × {B}))
2		rnxp 5051	. . . . 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 4793	. . . . . . 7 ⊢ (F:A–onto→{B} ↔ (F Fn A ∧ ran F = {B}))
7		fof 5269	. . . . . . 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 5452	. . . . . 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 4970	. . . . . . 7 ⊢ (F = ∅ ↔ ran F = ∅)
14	13	a1i 10	. . . . . 6 ⊢ (¬ B ∈ V → (F = ∅ ↔ ran F = ∅))
15		snprc 3788	. . . . . . . . . 10 ⊢ (¬ B ∈ V ↔ {B} = ∅)
16	15	biimpi 186	. . . . . . . . 9 ⊢ (¬ B ∈ V → {B} = ∅)
17	16	xpeq2d 4808	. . . . . . . 8 ⊢ (¬ B ∈ V → (A × {B}) = (A × ∅))
18		xp0 5044	. . . . . . . 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 2516  Vcvv 2859  ∅c0 3550  {csn 3737   × cxp 4770  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779

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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795

This theorem is referenced by: (None)