Theorem ofrn2 32537

Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Hypotheses

Ref	Expression
ofrn.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofrn.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
ofrn.3	⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)
ofrn.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
ofrn2	⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Proof of Theorem ofrn2

Dummy variables 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofrn.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6671	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑎 ∈ 𝐴)
4		fnfvelrn 7034	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
5	2, 3, 4	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐹‘𝑎) ∈ ran 𝐹)
6		ofrn.2	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
7	6	ffnd 6671	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
8		fnfvelrn 7034	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
9	7, 3, 8	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐺‘𝑎) ∈ ran 𝐺)
10		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))
11		rspceov 7418	. . . . 5 ⊢ (((𝐹‘𝑎) ∈ ran 𝐹 ∧ (𝐺‘𝑎) ∈ ran 𝐺 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
12	5, 9, 10, 11	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
13	12	rexlimdvaa 3135	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
14	13	ss2abdv 4026	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
15		ofrn.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16		inidm 4186	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
17		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝐹‘𝑎))
18		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) = (𝐺‘𝑎))
19	2, 7, 15, 15, 16, 17, 18	offval 7642	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
20	19	rneqd 5891	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) = ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
21		eqid 2729	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	21	rnmpt 5910	. . 3 ⊢ ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))}
23	20, 22	eqtrdi 2780	. 2 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))})
24		ofrn.3	. . . . 5 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)
25	24	ffnd 6671	. . . 4 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
26	1	frnd 6678	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
27	6	frnd 6678	. . . . 5 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵)
28		xpss12 5646	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
29	26, 27, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
30		ovelimab 7547	. . . 4 ⊢ (( + Fn (𝐵 × 𝐵) ∧ (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵)) → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
31	25, 29, 30	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
32	31	eqabdv 2861	. 2 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
33	14, 23, 32	3sstr4d 3999	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3911   ↦ cmpt 5183   × cxp 5629  ran crn 5632   “ cima 5634   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633

This theorem is referenced by:  sibfof  34304