Theorem ofrn2 32922

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 6704	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑎 ∈ 𝐴)
4		fnfvelrn 7073	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
5	2, 3, 4	syl2an2r 697	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐹‘𝑎) ∈ ran 𝐹)
6		ofrn.2	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
7	6	ffnd 6704	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
8		fnfvelrn 7073	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
9	7, 3, 8	syl2an2r 697	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐺‘𝑎) ∈ ran 𝐺)
10		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))
11		rspceov 7457	. . . . 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 1396	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
13	12	rexlimdvaa 3173	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
14	13	ss2abdv 4027	. 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 4187	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
17		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝐹‘𝑎))
18		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) = (𝐺‘𝑎))
19	2, 7, 15, 15, 16, 17, 18	offval 7681	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
20	19	rneqd 5926	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) = ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
21		eqid 2769	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	21	rnmpt 5945	. . 3 ⊢ ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))}
23	20, 22	eqtrdi 2820	. 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 6704	. . . 4 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
26	1	frnd 6712	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
27	6	frnd 6712	. . . . 5 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵)
28		xpss12 5674	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
29	26, 27, 28	syl2anc 595	. . . 4 ⊢ (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
30		ovelimab 7586	. . . 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 595	. . 3 ⊢ (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
32	31	eqabdv 2902	. 2 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
33	14, 23, 32	3sstr4d 4000	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095   ⊆ wss 3913   ↦ cmpt 5193   × cxp 5657  ran crn 5660   “ cima 5662   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408   ∘_f cof 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672

This theorem is referenced by:  sibfof  34671