Theorem ovelrn 7543

Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Assertion

Ref	Expression
ovelrn	⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))

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

Proof of Theorem ovelrn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnov 7540	. . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})
2	1	eleq2d 2822	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}))
3		ovex 7400	. . . . . 6 ⊢ (𝑥𝐹𝑦) ∈ V
4		eleq1 2824	. . . . . 6 ⊢ (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
5	3, 4	mpbiri 258	. . . . 5 ⊢ (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
6	5	rexlimivw 3134	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
7	6	rexlimivw 3134	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V)
8		eqeq1 2740	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
9	8	2rexbidv 3202	. . 3 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))
10	7, 9	elab3 3629	. 2 ⊢ (𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))
11	2, 10	bitrdi 287	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061  Vcvv 3429   × cxp 5629  ran crn 5632   Fn wfn 6493  (class class class)co 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-ov 7370

This theorem is referenced by:  efgredlem  19722  efgcpbllemb  19730  gsumval3  19882  lecldbas  23184  blrnps  24373  blrn  24374  qdensere  24734  tgioo  24761  xrge0tsms  24800  ioorf  25540  ioorinv  25543  ioorcl  25544  dyaddisj  25563  dyadmax  25565  mbfid  25602  ismbfd  25606  hhssnv  31335  xrge0tsmsd  33134  iccllysconn  35432  rellysconn  35433  icoreelrnab  37670  relowlssretop  37679  relowlpssretop  37680  islptre  46049