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Theorem ofeqd 7624
Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
Hypothesis
Ref Expression
ofeqd.1 (𝜑𝑅 = 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.
StepHypRef Expression
1 ofeqd.1 . . . . 5 (𝜑𝑅 = 𝑆)
21oveqd 7379 . . . 4 (𝜑 → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 5212 . . 3 (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpoeq3dv 7441 . 2 (𝜑 → (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥)))))
5 df-of 7622 . 2 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 7622 . 2 f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2802 1 (𝜑 → ∘f 𝑅 = ∘f 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3448  cin 3914  cmpt 5193  dom cdm 5638  cfv 6501  (class class class)co 7362  cmpo 7364  f cof 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-iota 6453  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  ofeq  7625  psrval  21333  resspsradd  21401  fedgmullem1  32364  fedgmullem2  32365  sitmval  32989  ldualset  37616  mendval  41539
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