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Theorem ofrn2 30384
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 6496 . . . . . 6 (𝜑𝐹 Fn 𝐴)
3 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑎𝐴)
4 fnfvelrn 6829 . . . . . 6 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
52, 3, 4syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐹𝑎) ∈ ran 𝐹)
6 ofrn.2 . . . . . . 7 (𝜑𝐺:𝐴𝐵)
76ffnd 6496 . . . . . 6 (𝜑𝐺 Fn 𝐴)
8 fnfvelrn 6829 . . . . . 6 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
97, 3, 8syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐺𝑎) ∈ ran 𝐺)
10 simprr 772 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑧 = ((𝐹𝑎) + (𝐺𝑎)))
11 rspceov 7185 . . . . 5 (((𝐹𝑎) ∈ ran 𝐹 ∧ (𝐺𝑎) ∈ ran 𝐺𝑧 = ((𝐹𝑎) + (𝐺𝑎))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
125, 9, 10, 11syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
1312rexlimdvaa 3277 . . 3 (𝜑 → (∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎)) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
1413ss2abdv 4028 . 2 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
15 ofrn.4 . . . . 5 (𝜑𝐴𝑉)
16 inidm 4178 . . . . 5 (𝐴𝐴) = 𝐴
17 eqidd 2825 . . . . 5 ((𝜑𝑎𝐴) → (𝐹𝑎) = (𝐹𝑎))
18 eqidd 2825 . . . . 5 ((𝜑𝑎𝐴) → (𝐺𝑎) = (𝐺𝑎))
192, 7, 15, 15, 16, 17, 18offval 7399 . . . 4 (𝜑 → (𝐹f + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
2019rneqd 5789 . . 3 (𝜑 → ran (𝐹f + 𝐺) = ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
21 eqid 2824 . . . 4 (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2221rnmpt 5808 . . 3 ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))}
2320, 22syl6eq 2875 . 2 (𝜑 → ran (𝐹f + 𝐺) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))})
24 ofrn.3 . . . . 5 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
2524ffnd 6496 . . . 4 (𝜑+ Fn (𝐵 × 𝐵))
261frnd 6502 . . . . 5 (𝜑 → ran 𝐹𝐵)
276frnd 6502 . . . . 5 (𝜑 → ran 𝐺𝐵)
28 xpss12 5551 . . . . 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 7309 . . . 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 2953 . 2 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
3314, 23, 323sstr4d 3998 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wrex 3133  wss 3918  cmpt 5127   × cxp 5534  ran crn 5537  cima 5539   Fn wfn 6331  wf 6332  cfv 6336  (class class class)co 7138  f cof 7390
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-of 7392
This theorem is referenced by:  sibfof  31616
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