Theorem ofrn 33113

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 7586	. . 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 4172	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	2, 3, 4, 5, 5, 6	off 7697	. 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶)
8	7	frnd 6712	1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3899   × cxp 5653  ran crn 5656  ⟶wf 6529  (class class class)co 7414   ∘_f cof 7677

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679

This theorem is used by: (None)