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Theorem ofrn2 29992
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 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
21ffnd 6280 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
32adantr 474 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐹 Fn 𝐴)
4 simprl 789 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑎𝐴)
5 fnfvelrn 6606 . . . . . 6 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
63, 4, 5syl2anc 581 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐹𝑎) ∈ ran 𝐹)
7 ofrn.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
87ffnd 6280 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
98adantr 474 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐺 Fn 𝐴)
10 fnfvelrn 6606 . . . . . 6 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
119, 4, 10syl2anc 581 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐺𝑎) ∈ ran 𝐺)
12 simprr 791 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑧 = ((𝐹𝑎) + (𝐺𝑎)))
13 rspceov 6952 . . . . 5 (((𝐹𝑎) ∈ ran 𝐹 ∧ (𝐺𝑎) ∈ ran 𝐺𝑧 = ((𝐹𝑎) + (𝐺𝑎))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
146, 11, 12, 13syl3anc 1496 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
1514rexlimdvaa 3242 . . 3 (𝜑 → (∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎)) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
1615ss2abdv 3901 . 2 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
17 ofrn.4 . . . . 5 (𝜑𝐴𝑉)
18 inidm 4048 . . . . 5 (𝐴𝐴) = 𝐴
19 eqidd 2827 . . . . 5 ((𝜑𝑎𝐴) → (𝐹𝑎) = (𝐹𝑎))
20 eqidd 2827 . . . . 5 ((𝜑𝑎𝐴) → (𝐺𝑎) = (𝐺𝑎))
212, 8, 17, 17, 18, 19, 20offval 7165 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
2221rneqd 5586 . . 3 (𝜑 → ran (𝐹𝑓 + 𝐺) = ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
23 eqid 2826 . . . 4 (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2423rnmpt 5605 . . 3 ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))}
2522, 24syl6eq 2878 . 2 (𝜑 → ran (𝐹𝑓 + 𝐺) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))})
26 ofrn.3 . . . . 5 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
2726ffnd 6280 . . . 4 (𝜑+ Fn (𝐵 × 𝐵))
281frnd 6286 . . . . 5 (𝜑 → ran 𝐹𝐵)
297frnd 6286 . . . . 5 (𝜑 → ran 𝐺𝐵)
30 xpss12 5358 . . . . 5 ((ran 𝐹𝐵 ∧ ran 𝐺𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
3128, 29, 30syl2anc 581 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
32 ovelimab 7073 . . . 4 (( + Fn (𝐵 × 𝐵) ∧ (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵)) → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3327, 31, 32syl2anc 581 . . 3 (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3433abbi2dv 2948 . 2 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
3516, 25, 343sstr4d 3874 1 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  {cab 2812  wrex 3119  wss 3799  cmpt 4953   × cxp 5341  ran crn 5344  cima 5346   Fn wfn 6119  wf 6120  cfv 6124  (class class class)co 6906  𝑓 cof 7156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-of 7158
This theorem is referenced by:  sibfof  30948
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