Theorem fnfvor 32592

Description: Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7627. (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 6822	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2		fveq2 6822	. . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
3	1, 2	breq12d 5102	. 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 4174	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
9		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	5, 6, 7, 7, 8, 9, 10	ofrfval 7620	. . 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 3573	1 ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5089   Fn wfn 6476  ‘cfv 6481   ∘_r cofr 7609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ofr 7611

This theorem is referenced by:  mplvrpmrhm  33577