Theorem ofrn2 32661

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 6750	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑎 ∈ 𝐴)
4		fnfvelrn 7116	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
5	2, 3, 4	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐹‘𝑎) ∈ ran 𝐹)
6		ofrn.2	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
7	6	ffnd 6750	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
8		fnfvelrn 7116	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
9	7, 3, 8	syl2an2r 684	. . . . 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 7499	. . . . 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 1371	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
13	12	rexlimdvaa 3162	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
14	13	ss2abdv 4089	. 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 4248	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
17		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝐹‘𝑎))
18		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) = (𝐺‘𝑎))
19	2, 7, 15, 15, 16, 17, 18	offval 7725	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
20	19	rneqd 5963	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) = ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
21		eqid 2740	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	21	rnmpt 5982	. . 3 ⊢ ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))}
23	20, 22	eqtrdi 2796	. 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 6750	. . . 4 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
26	1	frnd 6757	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
27	6	frnd 6757	. . . . 5 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵)
28		xpss12 5715	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
29	26, 27, 28	syl2anc 583	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
30		ovelimab 7630	. . . 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 583	. . 3 ⊢ (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
32	31	eqabdv 2878	. 2 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
33	14, 23, 32	3sstr4d 4056	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ⊆ wss 3976   ↦ cmpt 5249   × cxp 5698  ran crn 5701   “ cima 5703   Fn wfn 6570  ⟶wf 6571  ‘cfv 6575  (class class class)co 7450   ∘_f cof 7714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-of 7716

This theorem is referenced by:  sibfof  34307