Theorem fnrnfv 5364

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 4903	. 2 ⊢ ran F = {y ∣ ∃x⟨x, y⟩ ∈ F}
2		fnop 5186	. . . . . . . 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 5359	. . . . . . 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 2639	. . . . 5 ⊢ (∃x ∈ A y = (F ‘x) ↔ ∃x ∈ A (F ‘x) = y)
11		df-rex 2620	. . . . 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 2467	. 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 2615  ⟨cop 4561  ran crn 4773   Fn wfn 4776   ‘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-fv 4795

This theorem is referenced by:  fvelrnb  5365  fniinfv  5372  dffo3  5422  fniunfv  5466  fnrnov  5605