Theorem fnrnov 5605

Description: The range of an operation expressed as a collection of the operation's values. (Contributed by set.mm contributors, 29-Oct-2006.)

Assertion

Ref	Expression
fnrnov	⊢ (F Fn (A × B) → ran F = {z ∣ ∃x ∈ A ∃y ∈ B z = (xFy)})

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

Proof of Theorem fnrnov

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnfv 5364	. 2 ⊢ (F Fn (A × B) → ran F = {z ∣ ∃w ∈ (A × B)z = (F ‘w)})
2		fveq2 5328	. . . . . 6 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (F ‘⟨x, y⟩))
3		df-ov 5526	. . . . . 6 ⊢ (xFy) = (F ‘⟨x, y⟩)
4	2, 3	syl6eqr 2403	. . . . 5 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (xFy))
5	4	eqeq2d 2364	. . . 4 ⊢ (w = ⟨x, y⟩ → (z = (F ‘w) ↔ z = (xFy)))
6	5	rexxp 4826	. . 3 ⊢ (∃w ∈ (A × B)z = (F ‘w) ↔ ∃x ∈ A ∃y ∈ B z = (xFy))
7	6	abbii 2465	. 2 ⊢ {z ∣ ∃w ∈ (A × B)z = (F ‘w)} = {z ∣ ∃x ∈ A ∃y ∈ B z = (xFy)}
8	1, 7	syl6eq 2401	1 ⊢ (F Fn (A × B) → ran F = {z ∣ ∃x ∈ A ∃y ∈ B z = (xFy)})

Syntax hints:   → wi 4   = wceq 1642  {cab 2339  ∃wrex 2615  ⟨cop 4561   × cxp 4770  ran crn 4773   Fn wfn 4776   ‘cfv 4781  (class class class)co 5525

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-csb 3137  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-iun 3971  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-fv 4795  df-ov 5526

This theorem is referenced by:  ovelrn  5608