Theorem fnfvor 32667

Description: Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7639. (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 6833	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2		fveq2 6833	. . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
3	1, 2	breq12d 5110	. 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 4178	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7632	. . 3 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
12	4, 11	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥)𝑅(𝐺‘𝑥))
13		fnfvor.5	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
14	3, 12, 13	rspcdva 3576	1 ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050   class class class wbr 5097   Fn wfn 6486  ‘cfv 6491   ∘_r cofr 7621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ofr 7623

This theorem is referenced by:  mplmulmvr  33683  mplvrpmrhm  33691