Theorem ofeq 6052

Description: Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Assertion

Ref	Expression
ofeq	⊢ (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Proof of Theorem ofeq

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 987	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆)
2	1	oveqd 5859	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 4073	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
4	3	mpoeq3dva 5906	. 2 ⊢ (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))))
5		df-of 6050	. 2 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 6050	. 2 ⊢ ∘𝑓 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2224	1 ⊢ (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Syntax hints:   → wi 4   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ∩ cin 3115   ↦ cmpt 4043  dom cdm 4604  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844   ∘_𝑓 cof 6048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-iota 5153  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by: (None)