Theorem ofrn 31851

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 7574	. . 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 4217	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 3, 4, 5, 5, 6	off 7684	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶)
8	7	frnd 6722	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3947   × cxp 5673  ran crn 5676  ⟶wf 6536  (class class class)co 7405   ∘_f cof 7664

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 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666

This theorem is referenced by: (None)