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Theorem ofrn2 30725
Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
ofrn.1 (𝜑𝐹:𝐴𝐵)
ofrn.2 (𝜑𝐺:𝐴𝐵)
ofrn.3 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
ofrn.4 (𝜑𝐴𝑉)
Assertion
Ref Expression
ofrn2 (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Proof of Theorem ofrn2
Dummy variables 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.1 . . . . . . 7 (𝜑𝐹:𝐴𝐵)
21ffnd 6567 . . . . . 6 (𝜑𝐹 Fn 𝐴)
3 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑎𝐴)
4 fnfvelrn 6922 . . . . . 6 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
52, 3, 4syl2an2r 685 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐹𝑎) ∈ ran 𝐹)
6 ofrn.2 . . . . . . 7 (𝜑𝐺:𝐴𝐵)
76ffnd 6567 . . . . . 6 (𝜑𝐺 Fn 𝐴)
8 fnfvelrn 6922 . . . . . 6 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
97, 3, 8syl2an2r 685 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐺𝑎) ∈ ran 𝐺)
10 simprr 773 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑧 = ((𝐹𝑎) + (𝐺𝑎)))
11 rspceov 7281 . . . . 5 (((𝐹𝑎) ∈ ran 𝐹 ∧ (𝐺𝑎) ∈ ran 𝐺𝑧 = ((𝐹𝑎) + (𝐺𝑎))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
125, 9, 10, 11syl3anc 1373 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
1312rexlimdvaa 3213 . . 3 (𝜑 → (∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎)) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
1413ss2abdv 3993 . 2 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
15 ofrn.4 . . . . 5 (𝜑𝐴𝑉)
16 inidm 4149 . . . . 5 (𝐴𝐴) = 𝐴
17 eqidd 2740 . . . . 5 ((𝜑𝑎𝐴) → (𝐹𝑎) = (𝐹𝑎))
18 eqidd 2740 . . . . 5 ((𝜑𝑎𝐴) → (𝐺𝑎) = (𝐺𝑎))
192, 7, 15, 15, 16, 17, 18offval 7498 . . . 4 (𝜑 → (𝐹f + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
2019rneqd 5824 . . 3 (𝜑 → ran (𝐹f + 𝐺) = ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
21 eqid 2739 . . . 4 (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2221rnmpt 5841 . . 3 ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))}
2320, 22eqtrdi 2796 . 2 (𝜑 → ran (𝐹f + 𝐺) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))})
24 ofrn.3 . . . . 5 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
2524ffnd 6567 . . . 4 (𝜑+ Fn (𝐵 × 𝐵))
261frnd 6574 . . . . 5 (𝜑 → ran 𝐹𝐵)
276frnd 6574 . . . . 5 (𝜑 → ran 𝐺𝐵)
28 xpss12 5583 . . . . 5 ((ran 𝐹𝐵 ∧ ran 𝐺𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
2926, 27, 28syl2anc 587 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
30 ovelimab 7407 . . . 4 (( + Fn (𝐵 × 𝐵) ∧ (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵)) → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3125, 29, 30syl2anc 587 . . 3 (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3231abbi2dv 2876 . 2 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
3314, 23, 323sstr4d 3964 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  {cab 2716  wrex 3064  wss 3882  cmpt 5151   × cxp 5566  ran crn 5569  cima 5571   Fn wfn 6395  wf 6396  cfv 6400  (class class class)co 7234  f cof 7488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-of 7490
This theorem is referenced by:  sibfof  32048
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