Theorem fnfvor 33272

Description: Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7731. (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 6905	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2		fveq2 6905	. . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
3	1, 2	breq12d 5119	. 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 4126	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7724	. . 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 3581	1 ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106   Fn wfn 6550  ‘cfv 6555   ∘_r cofr 7713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ofr 7715

This theorem is used by:  mplmulmvr  34238  mplvrpmrhm  34246