Theorem ofrn 32419

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 7587	. . 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 4215	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 3, 4, 5, 5, 6	off 7698	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶)
8	7	frnd 6725	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2099   ⊆ wss 3945   × cxp 5671  ran crn 5674  ⟶wf 6539  (class class class)co 7415   ∘_f cof 7678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7418  df-oprab 7419  df-mpo 7420  df-of 7680

This theorem is referenced by: (None)