Theorem fnrnfv 5365

Description: The range of a function expressed as a collection of the function's values. (Contributed by set.mm contributors, 20-Oct-2005.)

Assertion

Ref	Expression
fnrnfv	⊢ (F Fn A → ran F = {y ∣ ∃x ∈ A y = (F ‘x)})

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fnrnfv

Step	Hyp	Ref	Expression
1		dfrn3 4904	. 2 ⊢ ran F = {y ∣ ∃x⟨x, y⟩ ∈ F}
2		fnop 5187	. . . . . . . 8 ⊢ ((F Fn A ∧ ⟨x, y⟩ ∈ F) → x ∈ A)
3	2	ex 423	. . . . . . 7 ⊢ (F Fn A → (⟨x, y⟩ ∈ F → x ∈ A))
4	3	pm4.71rd 616	. . . . . 6 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
5		fnopfvb 5360	. . . . . . 7 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y ↔ ⟨x, y⟩ ∈ F))
6	5	pm5.32da 622	. . . . . 6 ⊢ (F Fn A → ((x ∈ A ∧ (F ‘x) = y) ↔ (x ∈ A ∧ ⟨x, y⟩ ∈ F)))
7	4, 6	bitr4d 247	. . . . 5 ⊢ (F Fn A → (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ (F ‘x) = y)))
8	7	exbidv 1626	. . . 4 ⊢ (F Fn A → (∃x⟨x, y⟩ ∈ F ↔ ∃x(x ∈ A ∧ (F ‘x) = y)))
9		eqcom 2355	. . . . . 6 ⊢ (y = (F ‘x) ↔ (F ‘x) = y)
10	9	rexbii 2640	. . . . 5 ⊢ (∃x ∈ A y = (F ‘x) ↔ ∃x ∈ A (F ‘x) = y)
11		df-rex 2621	. . . . 5 ⊢ (∃x ∈ A (F ‘x) = y ↔ ∃x(x ∈ A ∧ (F ‘x) = y))
12	10, 11	bitri 240	. . . 4 ⊢ (∃x ∈ A y = (F ‘x) ↔ ∃x(x ∈ A ∧ (F ‘x) = y))
13	8, 12	syl6bbr 254	. . 3 ⊢ (F Fn A → (∃x⟨x, y⟩ ∈ F ↔ ∃x ∈ A y = (F ‘x)))
14	13	abbidv 2468	. 2 ⊢ (F Fn A → {y ∣ ∃x⟨x, y⟩ ∈ F} = {y ∣ ∃x ∈ A y = (F ‘x)})
15	1, 14	syl5eq 2397	1 ⊢ (F Fn A → ran F = {y ∣ ∃x ∈ A y = (F ‘x)})

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ⟨cop 4562  ran crn 4774   Fn wfn 4777   ‘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-fv 4796

This theorem is referenced by:  fvelrnb  5366  fniinfv  5373  dffo3  5423  fniunfv  5467  fnrnov  5606