Theorem ofrn2 31860

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 6718	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑎 ∈ 𝐴)
4		fnfvelrn 7082	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
5	2, 3, 4	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐹‘𝑎) ∈ ran 𝐹)
6		ofrn.2	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
7	6	ffnd 6718	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
8		fnfvelrn 7082	. . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
9	7, 3, 8	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → (𝐺‘𝑎) ∈ ran 𝐺)
10		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))) → 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)))
11		rspceov 7455	. . . . 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 3156	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎)) → ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
14	13	ss2abdv 4060	. 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 4218	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
17		eqidd 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = (𝐹‘𝑎))
18		eqidd 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) = (𝐺‘𝑎))
19	2, 7, 15, 15, 16, 17, 18	offval 7678	. . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
20	19	rneqd 5937	. . 3 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) = ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
21		eqid 2732	. . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	21	rnmpt 5954	. . 3 ⊢ ran (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = ((𝐹‘𝑎) + (𝐺‘𝑎))}
23	20, 22	eqtrdi 2788	. 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 6718	. . . 4 ⊢ (𝜑 → + Fn (𝐵 × 𝐵))
26	1	frnd 6725	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
27	6	frnd 6725	. . . . 5 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵)
28		xpss12 5691	. . . . 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 7584	. . . 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 2867	. 2 ⊢ (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹∃𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
33	14, 23, 32	3sstr4d 4029	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070   ⊆ wss 3948   ↦ cmpt 5231   × cxp 5674  ran crn 5677   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408   ∘_f cof 7667

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669

This theorem is referenced by:  sibfof  33334