Theorem ofrn 32621

Description: The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-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
ofrn	⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)

Proof of Theorem ofrn

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

Step	Hyp	Ref	Expression
1		ofrn.3	. . . 4 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)
2	1	fovcdmda 7517	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐶)
3		ofrn.1	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		ofrn.2	. . 3 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
5		ofrn.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4174	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 3, 4, 5, 5, 6	off 7628	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶)
8	7	frnd 6659	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3897   × cxp 5612  ran crn 5615  ⟶wf 6477  (class class class)co 7346   ∘_f cof 7608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610

This theorem is referenced by: (None)