Theorem ofeqd 6277

Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)

Hypothesis

Ref	Expression
ofeqd.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
ofeqd	⊢ (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Proof of Theorem ofeqd

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

Step	Hyp	Ref	Expression
1		ofeqd.1	. . . . 5 ⊢ (𝜑 → 𝑅 = 𝑆)
2	1	oveqd 6075	. . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 4206	. . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
4	3	mpoeq3dv 6127	. 2 ⊢ (𝜑 → (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 6275	. 2 ⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 6275	. 2 ⊢ ∘𝑓 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2292	1 ⊢ (𝜑 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Syntax hints:   → wi 4   = wceq 1398  Vcvv 2815   ∩ cin 3213   ↦ cmpt 4176  dom cdm 4754  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060   ∘_𝑓 cof 6273

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-iota 5317  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275

This theorem is referenced by:  psrval  14940  lgseisenlem3  16071  lgseisenlem4  16072