Theorem fnfvor 32924

Description: Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7695. (Contributed by Thierry Arnoux, 15-Jan-2026.)

Hypotheses

Ref	Expression
fnfvor.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnfvor.2	⊢ (𝜑 → 𝐺 Fn 𝐴)
fnfvor.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fnfvor.4	⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺)
fnfvor.5	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
fnfvor	⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

Proof of Theorem fnfvor

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2		fveq2 6885	. . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
3	1, 2	breq12d 5127	. 2 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
4		fnfvor.4	. . 3 ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐺)
5		fnfvor.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		fnfvor.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
7		fnfvor.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		inidm 4187	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7688	. . 3 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
12	4, 11	mpbid 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥)𝑅(𝐺‘𝑥))
13		fnfvor.5	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14	3, 12, 13	rspcdva 3590	1 ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086   class class class wbr 5114   Fn wfn 6535  ‘cfv 6540   ∘_r cofr 7677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ofr 7679

This theorem is referenced by:  mplmulmvr  33899  mplvrpmrhm  33907