Theorem ofeq 6138

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 999	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆)
2	1	oveqd 5939	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 4124	. . 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 5986	. 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 6135	. 2 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 6135	. 2 ⊢ ∘𝑓 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2254	1 ⊢ (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156   ↦ cmpt 4094  dom cdm 4663  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924   ∘_𝑓 cof 6133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-iota 5219  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135

This theorem is referenced by: (None)